JUL  39  ]m 

A  Contribution  to  the  Pedagogy 
of  Arithmetic 


BY 

ERNEST  C.  McDOUGLE 

Clark  University 


A  DISSERTATION  SUBMITTED  TO  THE  FACULTY  OF 
CLARK  UNIVERSITY,  WORCESTER,  MASS.,  IN  PARTIAL 
FULFILLMENT  OF  THE  REQUIREMENTS  FOR  THE  DE- 
GREE OF  DOCTOR  OF  PHILOSOPHY,  AND  ACCEPTED  ON 
THE  RECOMMENDATION  OF  WILLIAM  H.  BURNHAM. 


OP  THE 

UNIVERSITY 


Reprinted  from  the  Pedagogical  Seminary 
June,  1914,  Vol.  XXI,  pp.  161-218 


A  Contribution  to  the  Pedagogy 
of  Arithmetic 


BY 

ERNEST  C.  McDOUGLE 

Clark  University 


A  DISSERTATION  SUBMITTED  TO  THE  FACULTY  OF 
CLARK  UNIVERSITY,  WORCESTER,  MASS.,  IN  PARTIAL 
FULFILLMENT  OF  THE  REQUIREMENTS  FOR  THE  DE- 
GREE OF  DOCTOR  OF  PHILOSOPHY,  AND  ACCEPTED  ON 
THE  RECOMMENDATION  OF  WILLIAM  H.  BURNHAM. 


Reprinted  from  the  Pedagogical  Seminary 
June,  1914,  Vol.  XXI,  pp.  161-218 


r\  0^* 


»  • «  •'  « 


A  CONTRIBUTION  TO  THE  PEDAGOGY  OF 
ARITHMETIC* 


By  Ernest  C.  McDougle, 
Clark  University 


I.  Introduction 

Nearly  three  centuries  ago  Comenius  undertook  to  give  a 
method  to  teaching.  At  the  present  time  there  is  a  wide- 
spread movement  through  Experimental  Education  to  estab- 
lish it  upon  a  sound  scientific  basis.  Much  that  now  finds 
acceptance  in  both  curriculum  and  didactic  procedure  has 
found  a  place  through  conservative  respect  to  traditional 
philosophies  and  dogmas  of  the  past,  coupled  with  some  later 

*  It  is  fitting  that  I  should  here  acknowledge  my  indebtedness  to 
Dr.  G.  Stanley  Hall,  President  of  Gark  University,  for  suggesting 
the  field  of  investigation  which  led  to  this  thesis,  and  for  helpful 
direction  in  its  development;  to  Dr.  William  H.  Bumham,  Professor 
of  Pedagogy  and  School  Hygiene,  for  his  many  favors  in  suggest- 
ing material  to  be  embraced  in  the  research  upon  the  thesis  and 
for  valuable  criticism  of  the  treatment  of  the  topics  included;  to 
Eh".  Louis  N.  Wilson,  Librarian  of  Clark  University,  for  his  un- 
sparing pains  in  procuring  special  documents  for  my  use  in  gather- 
ing data;  to  Dr.  S.  A.  Courtis,  of  Detroit,  Mich.,  for  the  loan  of 
some  private  materials;  and  to  Miss  Rose  A.  Carrigan,  of  the 
Boston  Normal  School,  whose  kindness  made.it  possible  for  me  to 
witness  the  recent  tests  of  the  Boston  school  children  and  to  be 
furnished  with  additional  statistical  facts. 

I  wish  also  to  thank  the  following  persons  for  prompt  response 
to  my  personal  letters  asking  about  material  to  be  included  in  my 
survey:  Dr.  E.  L.  Thorndike,  of  Columbia  University;  Dr.  G. 
Deutchler,  Tubingen,  Germany;  Dr.  A.  W.  Stamper,  Chico,  Cal.; 
Dr.  W.  H.  Maxwell,  Superintendent  New  York  City  schools;  Mrs. 
Adelia  R.  Hornbrook,  San  Jose,  Cal.;  Prof.  J.  E.  Calfee,  Berea, 
Ky.;   and  Prof.   C.  H.  Dietrich,  Winchester,  Ky. 


162     CONTRIBX)frON  tb*THE  PEDAGOGY  OF  ARITHMETIC 

empirical  and  pragmatic  considerations.  The  present  critical 
studies  and  experiments  are  concerned  with  both  the  subjects 
in  the  curricula  and  the  methods  of  instruction.  In  these  re- 
searches many  laboratories  have  been  busy,  many  investiga- 
tors have  been  active,  and  much  helpful  work  has  been  done. 
It  is  now  necessary  that  the  results  should  be  brought  to- 
gether and  put  into  usable  form.  Experimental  Psychology 
has  been  too  busy  in  exploring  many  fertile  fields,  as  yet, 
to  give  attention  to  the  full  bearing  of  its  discoveries  upon 
Didactics,  while  Experimental  Pedagogy  is  frequently  too 
empirical  to  be  scientific.  For  these  reasons  it  is  essential 
that  the  synthesist  should  bring  together  the  modern  needs 
of  Pedagogy  and  the  contributions  of  Experimental  Psychol- 
ogy germane  to  the  processes  of  learning,  and  state  in  clear 
and  simple  language  the  norms  of  method  so  the  average 
teacher,  who  possesses  little  or  no  technical  nomenclature  of 
the  psychological  laboratory,  or  even  of  the  experimental 
pedagogist,  may  find  assistance  in  the  daily  routine  of  school 
duties. 

It  is  the  purpose  of  this  research  to  bring  together  many 
of  the  recent  tests  and  experiments  in  Arithmetic  and,  in 
connection  with  conclusions  drawn  from  them  bearing  upon 
better  texts  and  methods,  to  evaluate  them  for  practical  use 
in  the  regular  school  work. 

II.  Brief  Historic  Sketch  of  Arithmetic 

Arithmetic  is  the  oldest  science  developed  by  man.  As  an 
art  it  runs  much  farther  back  into  antiquity.  Its  first  use 
is  so  remote  that  it  is  difficult  to  separate  the  mythical  from 
the  real.  Anthropological  investigations  have  brought  much 
helpful  material  to  light  and  the  historical  genesis  of  Arith- 
metic is  coming  to  be  better  understood. 

Callisthenes  found  in  Babylon,  in  331  B.  C,  when  Alex- 
ander the  Great  captured  the  city,  burned  brick  astronomical 
records  running  back  to  2234  B.  C.  These  were  sent  to 
Aristotle,  according  to  Porphyry.  In  Egypt  no  uncivilized 
state  of  society  has  been  found.  Their  oldest  mathematical 
books  date  back  to  3400  B.  C,  although  Josephus  (62,  chap. 
7j  P-  50)  gives  Abraham  credit  for  introducing  Arithmetic 
into  Egypt,  when  he  says: 

"  He  communicated  to  them  Arithmetic  and  delivered  to  them 
the  science  of  Astronomy,  for  before  Abram  came  into  Egypt  they 
were  unacquainted  with  those  parts  of  learning,  and  that  science 
came  from  the  Chaldeans  into  Egypt,  and  from  thence  to  the 
Greeks  also." 

Among  the  Greeks  and  Romans,  as  well  as  among  most 


CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC     163 

primitive  peoples,  systems  of  counting  were  clumsy  and  im- 
perfect, and  it  became  necessary  to  invent  symbols  for  num- 
bers and  systems  of  numeration  for  the  practical  use  of 
these  symbols.  Counting-boards  and  the  abacus  v^ere  early 
invented  and  have  persisted  down  to  the  present  time.  We 
find  the  Chinese  and  Japanese  (64,  179)  using  the  swan-pan 
and  the  soroban  very  generally  even  to-day.  It  is  of  historic 
interest  to  note  that  the  first  printed  Arithmetic,  published 
at  Treviso,  near  Venice,  was  entitled  "  The  Art  of  the  Abacus 
for  Arithmetic."  Because  of  their  awkward  number  symbols 
the  Romans  used  calculating-boards  for  computations  and 
employed  their  symbols  only  in  stating  results.  The  process 
of  calculation  derives  its  name  from  the  Latin,  calculus, 
"  pebble,"  since  pebbles  were  used  as  counters  by  many  people 
around  the  Mediterranean. 

The  so-called  Arabic  characters  are  more  properly  Hindu. 
Leonardo  of  Pisa,  in  1200  A.  D.,  obtained  them  from  the 
Moors  and  two  years  later  he  published  a  system  of  com- 
putation using  them.  As  he  had  obtained  the  characters  from 
the  Arabs  they  received  the  name  Arabic.  They  have  been 
traced,  however,  to  the  Hindus.  After  their  introduction 
into  Europe,  a  long  contest  ensued  between  the  abacists  and 
the  algorists,  but  the  Hindu  system  gradually  spread  over 
the  continent  and  was  well  known  by  1400  A.  D.  Merchants 
discarded  the  Roman  notation  in  1550  and  the  monasteries 
and  colleges  followed  a  century  later.  With  the  new  system, 
Florentine  traders  and  writers  developed  double-entry  book- 
keeping and  worked  out  seven  operations :  Numeration,  Addi- 
tion, Subtraction,  Multiplication,  Division,  Involution,  and 
Evolution,  while  Italian  and  English  arithmeticians  simplified 
the  processes.  The  Arabs  added  from  left  to  right.  Garth, 
an  Englishman,  devised  a  plan  to  add  from  right  to  left. 
One  has  only  to  compare  the  solution  of  a  problem  in 
Division  by  Pacioli  or  Tartaglia  with  the  work  done  even 
in  the  third  grade  of  the  schools  to-day  to  note  what  simplifi- 
cation has  taken  place. 

The  invention  of  printing  with  movable  types,  together  with 
the  great  commercial  activity  carried  on  through  the  Han- 
seatic  League  and  other  agencies  gave  a  remarkable  stimulus 
to  algoristic  Arithmetic  in  Europe.  In  the  sixteenth  century 
sweeping  transitions  occurred.  The  manuscript  Arithmetics 
were  replaced  by  printed  books;  Roman  symbols  yielded  to 
the  Hindu  characters ;  the  Arithmetic  of  the  learned  became 
the  possession  of  the  common  people;  and  counters  dis- 
appeared in  favor  of  figures.  Many  authors  were  found  in 
Italy,  France,  Germany,  Holland,  and  England,  and  the  art 
of  reckoning  with  the  pen  rapidly  replaced  the  art  of  calcu- 


164     CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC 

lating  with  the  abacus.  The  Hanseatic  League  established 
Rechenschulen  along  its  trade  routes  and  these  commercial 
schools  had  much  to  do  in  keeping  Arithmetic  out  of  the 
regular  schools.  The  first  German  Arithmetic  appeared  in 
1482  from  the  pen  of  Ulrich  Wagner,  a  Rechenmeister  of  this 
League.  Many  topics,  such  as  Partnership  with  Time,  came 
into  use  then  as  a  practical  business  subject,  and  have  been 
retained  in  our  modern  text-books  long  after  the  business 
world  has  discarded  the  methods  they  present.  It  was 
through  the  influence  of  Pestalozzi  that  Arithmetic  was  given 
such  a  prominent  place  in  the  schools.  Despite  strenuous 
efforts  to  discredit  its  value  and  to  minimize  its  standing 
among  the  branches  of  learning  in  recent  years,  it  is  still 
receiving  from  12%  to  26%  of  the  entire  time  devoted  to 
recitation  in  the  Elementary  Schools  throughout  the  civilized 
countries  of  the  world   (90). 

III.  Psychology  of  Arithmetic 

The  big  question  among  teachers  is  to  know  how  to  teach 
according  to  sound  principles.  As  with  other  subjects  in 
the  curriculum,  scientific  methods  in  Arithmetic  must  ulti- 
mately be  based  upon  genetic  psychology.  Until  we  approach 
from  the  lower  side  the  many  questions  of  material  and 
method,  there  will  be  only  an  approximation  to  the  real 
solution.  Most  methods  have  been  superimposed,  so  to  speak, 
upon  the  child  from  above  and  only  in  late  years  has  there 
come  any  decided  scientific  tendency  to  study  the  genesis  of 
number  and  the  processes  of  computation  from  the  child's 
point  of  view.  These  studies  are  yet  confessedly  few  and 
do  not  warrant  an  attempt  at  a  full  statement  of  scientific 
method  based  upon  them.  What  has  been  learned  may  assist 
in  better  pedagogical  procedure  and  incite  to  further  original 
research. 

Genesis  of  Number  Ideas. — Major  (75,  167)  observed  his 
son  among  other  things  for  the  rise  of  his  ideas  of  number. 
He  found  him  able  to  miss  one  ball  out  of  his  wagon,  when 
three  were  in  it,  at  the  age  of  21  months.  While  the  child 
had  a  confused  idea  of  3,  4,  and  5,  at  three  years  of  age,  Major 
received,  many  times,  the  correct  number,  i,  or  2,  or  3,  when 
apples  were  used,  by  throwing  them  on  the  ground  and 
asking  for  a  certain  number.  Later,  the  child's  interest 
declined.  Preyer's  boy  missed  one  of  his  10  toys  (75,  166), 
at  the  age  of  only  18  months,  and  at  878  days  of  age  counted 
his  nine-pins  by  standing  them  in  a  row  and  saying :  "  Eins ! 
Eins !  Eins !  Noch  Eins !  Noch  Eins !  "  and  so  on  to  the  end 
of  the  row.     On  her  584th  day,  Dearborn's  little  daughter 


CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC     165 

(32a,  176)  counted  6  cattle  in  a  picture  by  saying:  "One, 
two;  one,  two;  one,  two."  He  thinks  she  knew  the  number 
2  as  early  as  the  543rd  day,  and  also  says  the  same  thing 
about  I.  Decroly  thinks  his  little  girl  loiew  2  before  she 
recognized  what  i  is  (33a,  119).  The  2  seemed  to  remain 
for  some  time  as  the  only  number  the  child  grasped.  Binet's 
daughter  at  30  months  of  age  comprehended  2,  and  could 
get  the  idea  of  4  at  51  months  (33a).  Lindner's  son  at  23 
months  had  the  number  2  (33a).  Moore's  three  children 
had  the  idea  of  2  at  22,  26,  and  29  months;  and  a  good 
idea  of  3  at  32,  and  53  months.     Scupin's  child  (33a)  knew 

2  at  22  months. 

Clara  and  William  Stern  studied  their  children  very  care- 
fully and  report  that  they  could  use  numbers  correctly  in 
connection  with  apples,  for  instance,  long  before  the  abstract 
idea  of  number  arose,  or  even  before  number  could  be  rightly 
applied  to  other  objects  of  less  interest  (33a).  Ordinals 
were  learned  before  cardinals.  Major  reports  ordinals  and 
cardinals  as  confusing  to  his  child  (75,  173),  while  Hilde 
Stern  could  get  the  fifth  finger,  but  did  not  seem  to  under- 
stand the  sum  of  her  fingers  and  thumb  on  one  hand  (74a). 
Decroly  made  many  careful  observations  upon  his  daughter 
and  contributes  a  very  interesting  opinion  that  she  knew  the 
number  2  long  before  she  had  the  idea  of  i.  This  does 
not  seem  to  have  been  observed  by  others  who  have  studied 
the  genesis  of  number  in  infants  and  deserves  to  have  more 
attention.  While  his  child  got  the  idea  of  two  at  19  months 
of  age,  she  was  able  to  differentiate  three,  at  28  months,  from 
two  or  one.  In  another  month  she  picked  out  2  objects  by 
the  aid  of  her  fingers,  and  at  35  months  did  this  without  using 
the  fingers.     At  41  months,  she  seemed  to  have  the  idea  of 

3  quite  well  in  mind  and  in  another  month  the  idea  of  4 
appeared  to  be  somewhat  clear.  Still,  as  many  other  child- 
observers  have  found  in  their  studies,  there  was  a  reversion 
of  both  interest  and  apparent. ability  later,  and  at  46  months 
of  age  the  little  girl  had  a  confused  idea  of  3  and  4,  calling 
them  simply,  *' more."  At  51  months,  3  arose  again  to  clear- 
ness and  soon  was  well  comprehended.  By  her  57th  month 
she  was  able  to  hand  the  correct  number  of  objects,  up  to  5, 
indicated  by  having  held  up  for  her  problem  one  or  more 
fingers.  This  showed  an  ability  to  abstract  the  idea  of  num- 
ber, say  from  four  fingers  held  up  by  her  father,  and  of 
applying  the  number  thus  gained  to  the  apples  or  other 
objects  asked  for.  Sully  found  his  child  at  4  years  3  months 
calling  big  beads  "  6,"  smaller  ones,  '*  5,"  and  still  smaller 
ones,  "4."  At  5  years  old,  however,  he  placed  four  crab- 
apples  upon  the  sand,  added  two  to  them  mentally,  and  begin- 


166     CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC 

ning  with  calling  the  first  one  ''  3/'  counted  them  correctly 
up  to  6.  He  could  also  count  his  four  playthings,  two  dolls, 
a  tin  soldier,  and  a  shell,  from  memory,  after  they  had  been 
put  away.  Sully  says  (121,  352)  that  the  drawings  of 
children  5  years  of  age  show  small  regard  for  the  five  fingers 
of  the  hand.  In  the  growth  of  infants  in  number  ideas  we 
find  marked  individual  differences,  and  it  can  only  be  said 
yet  that  the  idea  of  number  seems  to  be  forming  intelligently 
about  the  fourth  year  (75,  165). 

Lietzmann  {J2,  22)  thinks  schools  often  make  a  mistake 
in  proceeding  as  if  the  child  has  no  ideas  of  number  when 
it  enters  school.  "  The  beginnings  of  number  lie  much 
earlier.  When  the  child  comes  to  school,  it  will,  in  a  majority 
of  cases,  already  know  the  number  words  possibly  up  to 
10,  or  12."  Of  1,217  children  entering  the  Volksschulen  in 
Breslau,  10%  could  count  up  to  5,  78%  could  count  up  to 
10,  4%  up  to  100,  and  nearly  1%  over  100. 

Ballard  says  (9,  58)  :  "  It  is  a  well  known  fact  that  chil- 
dren learn  to  count  of  their  own  accord.  They  do  it  at 
home,  on  the  playground,  and  at  their  games.  It  is  impossible 
to  stop  them."  "  I  tested  a  school  where  no  counting  is 
especially  taught  and  found  that  about  60%  of  the  children 
could  count  up  to  20,  before  they  were  5  years  old,  and  about 
30%  could  count  up  to  30  before  they  were  6."  Meumann 
(79,  Bd.  2,  345)  agrees  with  Preyer  and  the  other  few  child 
psychologists  that  the  child  has  a  somewhat  fully  intelligible 
use  of  numbers  from  i  to  10  at  the  earliest  toward  the 
close  of  the  4th  year  of  its  life.  In  his  report  of  the  Fielden 
School,  Manchester,  Eng.,  Harrison  (54,  269)  says  the  chil- 
dren play  with  dominoes  and  "  the  result  is,  that  already  the 
five-year  olds  are  able  to  write  correctly  on  the  blackboard  the 
result  of  2  plus  7,  9  plus  5,  etc."  This  agrees  with  Mon- 
tessori  who  has  children  at  three  to  begin  counting  with  but- 
tons, plates,  or  money,  and  later  with  sets  of  blocks. 

There  are  two  very  distinct  schools  of  educators  on  the 
question  of  the  origin  of  the  child's  notion  of  number.  One 
party  maintains  that  ideas  of  number  are  developed  through 
the  simultaneous  perception  of  several  objects,  or  stimuli, 
presented  to  the  senses.  With  Newcomb  (87)  they  hold 
that  "  our  teaching  of  numbers  is  too  abstract, — too  much 
dissociated  from  objects  of  sense."  Many  experiments  have 
been  performed  in  recent  years  to  determine  how  many 
objects  one  may  be  able  to  apprehend  accurately  without 
counting.  For  this  purpose  the  time  exposure  is  made 
as  brief  as  possible  so  as  to  prevent  counting.  Nanu 
(86)  used  bright  dots  on  a  dark  background  and  gave 
a    time    exposure    of    33/1000    of    a     second.       She    ar- 


CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC     167 

ranged  the  dots  in  different  figures  and  found  that  5  could 
be  perceived  in  a  line,  10  in  a  parallelogram,  and  8  in  a 
hexagon  in  75%  of  the  cases.  Lay  (67)  performed  many- 
experiments  and  found  a  greater  percentage  of  successes  at- 
tended the  arrangement  of  dots  in  quadrate  form.  He  is  a 
strong  advocate  of  the  objective  method  of  teaching  the  early 
number  work.  He  quotes  the  experiments  of  Goldscheider 
and  Miiller,  von  Scheele,  Schneider,  Kiilpe,  Cattell,  Dietze, 
Warren  and  Messenger  in  support  of  his  claims.  However, 
his  contentions  are  not  without  vigorous  criticism  from  Walse- 
mann,  Knilling,  Knoche,  and  the  whole  school  of  Herbartians. 
McLellan  and  Dewey  (76)  would  base  the  development  of 
number  upon  measuring, — upon  the  ratio  idea.  They  hold 
that  there  is  no  number  without  measurement  (p.  242)  nor 
measurement  without  the  fraction  implied.  Tear  reviewed 
the  Speer  arithmetic  a  half  generation  ago  and  quotes  Newton 
(123,  631)  :  "  Number  is  the  abstract  ratio  of  one  quantity  to 
another  quantity  of  the  same  kind."  So,  also,  the  great 
Swiss  mathematician,  Euler,  is  cited :  "  Number  is  the  ratio 
of  one  quantity  to  another  quantity  taken  as  a  unit." 

In  direct  opposition  to  this  notion  of  the  origin  of  number 
in  the  child  mind  and  to  the  consequent  procedure  in  teach- 
ing, there  is  a  strong  party  which  maintains  that  the  presen- 
tation of  the  objects  in  a  series,  or  the  stimuli  in  succession, 
is  the  proper  method.  Gilbert  (43,  no)  believes  that  count- 
ing is  "  the  first  step  in  systematic  thinking,"  while  Phillips 
decides  (95,  22'j')  that  "  the  first  step  is  surely  the  forma- 
tion of  the  series-idea."  He  holds  that  counting  is  funda- 
mental, and  that  children  forbidden  to  count  on  their  fingers 
sometimes  count  by  using  their  toes,  or  move  an  elbow,  or 
press  a  muscle,  or  clear  the  throat  slightly  in  order  to  follow 
the  series.  Among  his  reasons  for  rejecting  the  Grube 
method,  Badanes  says  (6,  34)  :  "  It  is  false  from  the  point 
of  view  of  Arithmetic  as  a  science  and  as  an  art.  It  ignores 
the  process  of  counting."  The  dispute  really  carries  us  back 
into  the  philosophical  question  as  to  whether  number  has 
time-relation  only,  or  space-relation  only,  or  neither,  or  both. 
It  will  suffice  to  present  the  writer's  view  on  the  pedagogical 
significance  of  the  discussion  to  quote  with  approval  the  words 
of  Meumann  (79,  338)  :  From  the  psychological  point  of 
view  number-concepts  seem  to  possess  both  temporal  and 
spatial  qualities,  and  either  of  the  views  alone  represents 
necessarily  a  one-sided  notion,  that  by  the  series-method  and 
by  the  simultaneous-method  the  child  gains  something  of 
power  not  contained  in  the  other.  "  The  fact  that  by  both 
methods  good  results  can  be  obtained  shows  that  a  comple- 
ment of  these  methods,  or  the  simultaneous  employment  of 


168     CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC 

both,  must  be  the  right  way  of  complete  comprehension  of 
number  for  the  child." 

On  the  side  of  anthropology,  many  systems  of  notation 
have  been  found  among  primitive  peoples,  and  no  definite 
correlation  seems  to  exist  between  these  systems  and  the 
civil  development  of  the  tribes.  The  highly  civilized  Peru- 
vians knew  almost  no  arithmetic  as  an  art  and  nothing  of  it 
as  a  science,  according  to  Conant  (24,  150),  while  the  Yoru- 
bas,  a  very  barbarous  tribe  in  Africa  could  count  quite  ex- 
pertly. Many  savage  tribes  count  only  up  to  two,  and  have 
number  words  only  for  "  one,"  "  two,"  and  "  many  "  or  some 
other  verbal  device  for  distinguishing  their  first  definite 
number  ideas  from  the  indefinite  ones  lying  beyond.  The 
lowest  Brazilian  tribes  count  to  3,  and  the  Carribees,  Galibi, 
Abipones  and  many  others  go  up  to  4.  As  a  rule  the  South 
American  and  Australian  tribes  count  seldom  above  3  or  4. 
Among  some  of  the  Australians  only  binary  systems  prevail, 
while  ternary  and  quaternary  systems  abound  among  the 
Indian  tribes  of  South  America.  Yet,  some  Pacific  island 
tribes  have  been  found  with  ability  to  count  up  into  millions 
in  their  trade  in  fish  and  breadfruit.  Quinary  scales  are 
widely  diffused  throughout  the  world,  and  a  few  octonary 
systems  are  believed  to  have  existed.  Some  scant  traces  of 
vigesimal  systems  have  been  found.  All  these  facts  show 
how  slowly  and  imperfectly  the  concept  of  number  has  arisen 
among  primitive  peoples,  who  seem  to  have  used  numbers 
only  as  necessity  forced  the  matter  upon  them.  Need  for 
counting  in  their  barter  with  one  another  had  more  influence 
upon  number  than  did  their  general  intelligence,  or  any  sub- 
jective interest  in  it.    Conant  (25,  31)  says: 

"If  the  life  of  any  tribe  is  such  as  to  induce  trade  and  barter 
with  their  neighbors,  a  considerable  quickness  in  reckoning  will 
be  developed  among  them.  Otherwise  this  power  will  remain  dor- 
mant because  there  is  but  little  in  the  life  of  primitive  man  to  call 
for  its  exercise." 

More  recently  Boas  (10,  65-66)  has  stated  this  as  follows: 

"  The  fact  that  generalized  forms  of  expression  are  not  used  does 
not  prove  inability  to  form  them,  but  merely  proves  that  the  mode 
of  life  of  the  people  is  such  that  they  are  not  required;  that  they 
would,  however,  develop  just  as  soon  as  needed.  This  point  of  view 
is  also  corroborated  by  a  study  of  the  numeral  systems  of  primitive 
languages.  As  is  well  known,  many  languages  exist  in  which  the 
numerals  do  not  exceed  two  or  three.  It  has  been  inferred  from 
this  that  the  people  speaking  these  languages  are  not  capable  of 
forming  the  concept  of  higher  numbers.  I  think  this  interpretation 
of  the  existing  conditions  is  highly  erroneous.  Peoples  like  the 
South  American  Indians  (among  whom  these  defective  numeral 
systems  are  found),  or  like  the  Eskimo  (whose  old  system  of  num- 
bers probably  did  not  exceed  ten),  are  presumably  not  in  need  of 


CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC     169 

higher  numerical  expressions,  because  there  are  not  many  objects 
that  they  have  to  count.  On  the  other  hand,  just  as  soon  as  these 
same  people  find  themselves  in  contact  with  civilization,  and  when 
they  acquire  standards  of  value  that  have  to  be  counted,  they  adopt 
with  perfect  ease  higher  numerals  from  other  languages  and  develop 
a  more  or  less  perfect  system  of  counting.  ...  It  must  be  borne 
in  mind  that  counting  does  not  become  necessary  until  objects  are 
considered  in  such  generalized  form  that  their  individualities  are 
entirely  lost  sight  of.  For  this  reason  it  is  possible  that  even  a 
person  who  has  a  flock  of  domesticated  animals  may  know  them 
by  name  and  by  their  characteristics  without  ever  desiring  to  count 
them.  Members  of  a  war  expedition  may  be  known  by  name  and 
may  not  be  counted.  In  short,  there  is  no  proof  that  the  use  of 
numerals  is  in  any  way  connected  with  the  inability  to  form  con- 
cepts of  higher  numbers." 

It  may  not  be  an  unfair  or  unwarranted  deduction  from  all 
these  studies  and  views  to  believe  that  children  at  school 
entrance  in  America  may  have  ample  ability  to  delight  in 
numbers  although  they  may  show  little  interest  in  them  up 
to  that  time.  The  child-mind  no  doubt  expands  intelligently 
with  its  growth  in  experience  with  objects  of  multitude,  much 
as  the  development  of  the  primitive  mind  does,  as  described 
by  Boas.  Its  early  work  with  these  numbers  can  be  moti- 
vated and  made  attractive  in  a  manner  paralleling  the  race 
expansion.  "  The  child  is  a  natural  symbolist,"  says  Mary 
R.  Ailing- Aber  (3,  171).  "A  corn-cob  with  a  dress  on  it 
will  do  for  a  baby  and  a  stick  with  no  additions,  for  a  horse. 
To  let  one  thing  stand  for  another  is  as  easy  to  a  child  as 
to  breathe."  There  is  an  easy  transition  from  the  objects, 
too,  to  numbers  and  then  from  numbers  to  symbols  at  an 
age  corresponding  to  school  entrance. 

Time  of  Beginning  Arithmetic. — With  regard  to  the  time 
when  Arithmetic  should  be  introduced  into  the  schools  and 
when  it  should  be  completed  there  is  some  difference  of 
opinion,  and  the  matter  is  just  now  in  the  polemical  stage. 
One  group  of  educators  holds  that  it  should  not  be  taught, 
except  incidentally,  in  the  first  grade,  or  first  and  second,  or 
the  first  three  grades.  The  majority  report  of  the  Committee 
of  Fifteen  of  the  National  Educational  Association,  in  1895, 
urged  the  beginning  of  the  subject  in  the  second  and  its 
completion  in  the  sixth  grade.  Burnham  (17,  65)  believes 
there  is  "  ample  reason  for  postponing  the  work  of  Arithmetic 
until  the  age  of  10,  or,  more  accurately,  to  that  stage  of  devel- 
opment which  is  likely  to  be  found  in  normal  children  at 
this  age.  While  it  is  greatly  to  be  desired  that  more  investi- 
gations be  made  in  regard  to  this  subject,  with  our  present 
experience  this  seems  to  be  a  wise  rule."  Chartres  (22,  278) 
says  that  "  separate  Arithmetic  classes  should  not  be  taught 
in  the  first  grade;  it  is  better  to  defer  them  to  the  second 


170     CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC 

grade,  and  probably  it  would  be  better  to  begin  Arithmetic 
in  the  fourth  grade  if  our  text-books  were  built  with  that  in 
view."  So  also  writes  Stamper  (113,  258):  "The  general 
tendency  in  this  country  is  to  refrain  from  all  drill  work  in 
the  first  year.  In  fact,  some  schools  defer  such  work  until 
the  middle  of  the  second  year,  or  the  beginning  of  the  third." 
On  the  other  side  of  the  controversy  are  to  be  found  many 
ardent  protagonists.  Smith  (105,  128)  says:  "Not  to  put 
Arithmetic  as  a  topic  in  the  first  grade  is  to  make  sure  that 
it  will  not  be  seriously  or  systematically  taught  there  in  nine- 
tenths  of  the  schools  of  the  country.  The  average  teacher, 
not  in  the  cities  merely,  but  throughout  the  country  generally, 
will  simply  touch  upon  it  in  the  most  perfunctory  way.  What- 
ever of  scientific  statistics  we  have  show  that  this  is  true, 
and  that  children  so  taught  are  not,  when  they  enter  the 
intermediate  grades,  as  well  prepared  in  Arithmetic  as  those 
who  have  studied  the  subject  as  a  topic  from  the  first  grade 
on."  In  a  very  recent  article  (106,  95)  he  further  argues: 
"All  the  talk  about  having  no  Arithmetic  in  grades  i  and  2, 
or  leaving  it  to  the  whim  of  the  teacher,  has  not  shaken  the 
belief  of  the  great  schools  of  the  world  in  the  wisdom  of 
Pestalozzi's  judgment."  "Arithmetic  is  a  game  and  all  boys 
and  girls  are  mere  players.  We  have  not  learned  this  very 
thoroughly  yet,  but  we  are  making  progress."  Montessori 
(83,  326)  claims  to  achieve  some  wonderful  results  even 
with  children  of  pre-school  age.  Greenwood  (47)  dissented 
from  the  report  of  the  Committee  of  Fifteen  and  furnished  a 
verbatim  report  of  some  actual  teaching  and  results  in  the 
Kansas  City  Schools  as  proof  of  his  contention  that  Arith- 
metic is  eminently  successful  in  the  lower  grades.  "  No 
greater  difficulty  to  get  small  children  to  grasp  the  nature 
of  a  fraction  as  such  than  in  getting  them  to  grasp  the  simple 
whole  numbers,  .  .  .  Children  get  the  idea  of  half, 
third,  quarter,  long  before  they  enter  school."  Hence,  he 
advocates  teaching  them  to  add,  subtract,  multiply  and  divide 
fractions  in  the  first  grade.  Cook  (7)  says:  "I  visited  the 
Kansas  City  schools  and  testify  that  Mr.  Greenwood  has  not 
overestimated  conditions.  I  took  some  third-grade  work  home 
and  tried  it  on  Normal  students,  and  they  couldn't  do  it  as 
rapidly  as  those  children  did  it."  In  Germany  a  little  more 
than  20%  of  the  time  in  the  first  two  grades  is  devoted  to 
Arithmetic.  In  the  United  States,  according  to  the  report 
formulated  in  191 1  by  the  American  Committee  of  the  Inter- 
national Commission  on  the  Teaching  of  Mathematics  (56, 
16-65,  75-7^)  cities  reporting  about  one-tenth  of  the  school 
population  of  the  country  show  that  Arithmetic  is  taught  as 
a  topic  in  the  first  grade  in  71.5%  ;  in  the  second  grade  it 


CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC     171 

is  introduced  in  22%  ;  and  in  the  third  grade  in  6.5%.  (See 
Table  E  for  further  details.)  In  country,  or  rural  schools, 
it  may  be  said  to  be  taught  in  the  first  grade  in  practically 
all  the  schools. 

The  writer  personally  believes  that  there  should  be  sys- 
tematic teaching  of  numbers  in  the  first  grades.  The  child's 
natural  interest  in  numbers  and  the  rich  opportunities  for  pre- 
senting numbers  in  concrete  objects  should  be  utilized.  Mere 
rote  work  must  be  avoided,  but  these  early  years  are  of 
inestimable  value  in  furnishing  a  substantial  foundation  in 
the  child's  individual  growth  in  the  comprehension  of  num- 
bers. Counting  may  also  be  used  in  these  grades  to  diversify 
the  work  and  add  to  the  useful  results. 

Utility  and  Discipline. — ^Another  question  of  much  interest 
and  one  upon  which  there  is  considerable  controversy  yet,  is 
that  of  the  so-called  disciplinary  value  of  Arithmetic.  The 
matter  is  at  least  as  old  as  Plato,  who  says  in  his  Republic: 
"And  you  have  further  remarked  that  those  who  have  a 
natural  talent  for  calculation  are  generally  quick  at  every 
other  kind  of  knowledge,  and  even  the  dull,  if  they  have 
arithmetical  training,  gain  in  quickness,  if  not  in  any  other 
way."  In  mathematics,  perhaps,  more  than  in  any  other 
subject,  the  doctrine  of  formal  discipline,  or  transfer  of  train- 
ing, has  been  most  successfully  maintained.  If  Arithmetic 
has  not  been  kept  in  the  curriculum  as  a  practical  subject  it 
has  staid  there  as  a  disciplinarian  of  the  intellect.  John 
Stuart  Mill  attributed  his  success  in  speculation  to  his  mathe- 
matical training, — "  the  habit  of  never  accepting  half  solu- 
tions of  difficulties  as  complete,  never  abandoning  a  puzzle, 
but  again  and  again  returning  to  it  until  it  was  cleared  up; 
never  allowing  obscurities  in  a  subject  to  remain  unexplained 
because  they  did  not  seem  important;  never  thinking  I  per- 
fectly understood  any  part  of  a  subject  until  I  understood 
the  whole."  This  is  not  the  place  to  discuss  the  general  ques- 
tion of  mental  discipline,  or  transfer  of  training,  upon  which 
an  extensive  and  varied  literature  has  been  produced  within 
the  past  decade.  It  is  not  well  to  accept  either  of  the  two 
extreme  positions  noted  in  the  literature,  but  at  present  the 
value  of  arithmetic  in  the  school  course  may  be  defended 
upon  both  practical  and  disciplinary  grounds.  In  its  utilitarian 
aspects  the  demands  of  every-day  commercial  life  are  suf- 
ficient proof.  And  on  the  subjective  side,  if  it  should  be 
shown  that  one  intellectual  trait  does  not  and  cannot  assist 
any  other,  still  the  Einstellung  toward  matters  under  con- 
sideration trained  into  children  in  Arithmetic  may  be  ad- 
vanced as  evidence  of  the  subjective  discipline  of  Arithmetic. 


172     CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC 

The  certainties  found  in  Arithmetic,  too,  have  a  moral  value 
to  children  as  they  approach  so  much  of  uncertainty  in  other 
studies. 

IV.  Experimental  Studies 

Many  experimental  studies  of  the  teaching  of  Arithmetic 
and  the  various  processes  of  learning  the  different  operations 
in  handling  numbers  have  been  made,  notably  within  the  past 
two  years.  While  they  have  been  concerned  principally  with 
only  the  fundamental  operations,  a  synthetic  study  of  their 
results  should  throw  some  added  light  upon  our  problem  of 
finding  a  scientific  basis  for  Arithmetical  Methods.  Thous- 
ands of  children  have  been  tested  and  drilled  under  more 
or  less  controlled  conditions,  and  the  results  are  now  becom- 
ing available  for  comparative  pedagogical  purposes.  Much 
yet  remains  to  be  learned  by  further  experimentation,  but  a 
consideration  of  those  studies  that  have  been  made  and  pub- 
lished will  assist  materially  in  determining  more  clearly  than 
has  been  done  heretofore  the  weakness  of  the  present  methods, 
their  strong  features,  and  suggest  the  next  step  in  the  search 
for  a  sound  pedagogy  of  Arithmetic. 

Influence  of  Puberty. — Voigt  made  some  studies  upon  chil- 
dren from  ten  to  fourteen  years  of  age  in  the  Volksschulen 
(80,  117).  Instead  of  using  the  ordinary  decimal  system 
of  notation,  he  employed  systems  in  which  8  and  6  were  the 
bases.  By  this  means  he  reduced  to  a  minimum  the  use  of 
knowledge  already  possessed  by  the  children.  Among  his 
results  are  these:  i.  The  learning  of  new  systems  does  not 
progress  gradually,  but  by  "  leaps ;"  2,  Between  the  ages  of 
13  and  14  the  boys  showed  a  marked  increase  in  ability;  3,. 
Between  the  ages  of  12  and  13  the  girls  showed  this  rise  in 
ability.  It  was  noticed  that  the  onset  of  puberty  gave  a 
decided  increase  in  the  ability  to  work  independently  in 
numbers.  Prior  to  this  time  the  children  as  a  rule  work 
mechanically,  according  to  "  copy."  The  boys  reached  the 
period  of  independent  work  from  i  to  i^  years  in  physiolog- 
ical age  later  than  the  girls ;  hence,  the  girls  of  the  same 
age  as  the  boys  after  the  beginning  of  puberty  are  usually 
more  than  one  year  ahead  of  them  in  number  work.  That 
is,  problems  which  boys  can  solve  independently  in  their 
eighth  year  in  school  can  be  as  easily  solved  by  girls  in  their 
seventh  year.  Rice  and  Courtis  both  found  the  6th  grade 
especially  troublesome,  as  there  were  disturbances  in  the 
scores  of  that  grade  in  the  many  records  which  they  gathered. 
In  the  recent  tests  in  Boston  (38,  23)  this  errancy  was  found 
in  the  7th  grade.     Since  Boston  admits  children  at  five  years 


CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC     173 

of  age,  the  7th  grade  there  would  correspond  to  the  6th  grade 
in  physiological  age  elsewhere.  It  appears  from  the  tests  of 
Voigt,  Rice  and  Courtis  that  the  dawn  of  adolescence,  affect- 
ing as  it  does  markedly  the  physiological  nature  of  the  child, 
has  also  a  great  effect  upon  its  ability  to  do  accurate  and 
independent  work  with  numbers.  Before  this  time,  mechan- 
ical work  is  done;  after  this  period  the  rise  of  independence 
and  self-reliance  changes  the  emphasis  to  the  reasoning  phases 
of  arithmetical  solutions. 

Value  of  Drill. — ^A  number  of  experimental  studies  have 
been  made  to  arrive  at  some  fundamental  facts  concerning 
the  value  of  drill.  Brown  (13,  8iff)  gave  the  Stone  tests  to 
6th,  7th,  and  8th  grades  of  the  practice  school  of  the  Eastern 
Illinois  State  Normal  School,  consisting  of  51  pupils,  18  boys 
si^d  33  girls.  Two  sections  were  made  of  them.  One  was 
drilled  upon  fundamentals  for  five  minutes  each  day  for  thirty 
days,  while  the  other  pursued  regular  work.  A  test  at  the 
end  showed  that  the  drilled  pupils  made  a  much  larger  ad- 
vancement than  the  others.  His  results  also  showed  that 
the  6th  grade  children  profited  more  from  drill  than  did  the 
next  higher  grades.  Later,  he  carried  his  experiments  into 
three  school  systems  (13,  488),  confining  the  tests  to  the  6th 
grade  and  to  the  four  fundamental  operations.  In  all,  he 
tested  222  pupils,  no  boys  and  112  girls.  Only  twenty  days 
were  allowed  between  the  first  and  the  second  tests,  during 
which  time  one  group  in  each  school  was  given  five  minutes 
extra  drill  in  the  fundamentals  besides  the  regular  work 
which  the  other  section  followed.  The  results  showed  a  gain 
on  the  part  of  the  non-drill  group  in  problems  solved  of  6.4%  ; 
in  Addition,  of  6.8%;  in  Subtraction,  of  11.9%;  in  Multipli- 
cation, of  10.9%;  in  Division,  of  15.4%.  The  drill  group 
gained  on  these  same  items  respectively:  16.9%,  18.5%,  32%, 
24.1%,  and  34.2%.  These  are  only  the  aggregates.  Of  the 
112  cases  of  drilled  pupils,  95  gained,  5  did  not  advance  and 
12  lost.  Of  the  no  cases  of  non-drilled  pupils,  50  gained, 
7  had  the  same  score  as  at  the  beginning,  and  53  lost.  In 
the  individual  studies  and  aggregates  submitted  it  appears  that 
the  drilled  groups  gained  from  two  to  three  times  as  much 
as  the  non-drilled  groups.  "All  teachers  of  the  drill  classes 
reported  an  improvement  also  in  the  text-book  work."  The 
drill  excited  them  to  keener  interest  in  the  regular  lessons 
(13,  489). 

In  March,  1912,  Phillips  (93)  gave  the  Stone  tests  to  33 
boys  and  36  girls  of  the  6th,  7th,  and  8th  grades.  Each  grade 
was  divided  into  two  groups,  one  pursuing  regular  work  and 


174     CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC 

the  other  receiving  in  addition  a  ten-minute  drill  daily  upon 
fundamental  operations  and  upon  reasoning  upon  mental 
problems.  At  the  end  of  two  months  the  two  groups  were 
again  submitted  to  the  Stone  test  with  the  following  results: 
6th  grade  non-drill,  gain  in  Fundamentals,  55%  ;  drill  group, 
45%  ;  7th  grade,  in  Fundamentals,  non-drill,  10%  ;  drill  group, 
22%  ;  8th  grade,  in  Fundamentals,  non-drill,  16%  ;  drill,  25%  ; 
all  grades,  gain  in  Fundamentals,  non-drill,  27%;  drill,  31%. 
In  Reasoning :  the  6th  grade  gain  was :  non-drill,  36%  ;  drill, 
55%;  7th  grade,  non-drill,  17%;  drill,  29%;  8th  grade,  non- 
drill,  12%;  drill,  15%;  all,  non-drill,  22%;  drill,  33%. 

Starch  (114)  gave  15  observers  eight  preliminary  tests, 
six  in  arithmetical  operations  and  two  in  auditory  memory 
span  for  numbers.  Of  these  observers,  8  were  then  given 
fourteen  days'  practice  in  mental  multiplication  of  50  prob- 
lems each  day,  totaling  700  problems.  The  other  7  observers 
were  given  no  practice.  On  the  second  test  the  practiced  ob- 
servers showed  from  twenty  to  forty  per  cent  improvement 
more  than  the  others  in  the  arithmetical  operations,  while 
there  was  little  change  in  the  memory  span  in  either  group. 
Thorndike  (125)  experimented  with  33  adults  to  learn  the 
increase  in  efficiency  in  mental  multiplication,  judged  by  the 
reduction  in  the  time  required.  He  used  no  figure  below  3 
and  none  was  repeated.  All  his  subjects  showed  improve- 
ment through  drill,  and  he  says: 

"  The  fact  that  these  mature  and  competent  minds  improved  in 
the  course  of  so  short  a  training  so  much  as  to  be  able  to  do  an 
equal  task  in  two-fifths  of  the  time  first  taken  is  worthy  of  atten- 
tion." "The  most  ardent  advocate  of  the  general  influence  of  specific 
practice  would  not,  I  judge,  claim  that  ten  hours  drill  in  any  one 
thing  could  improve  an  already  well-educated  adult  50%,  or  5%, 
or  even  1%  in  the  average  of  all  kis  intellectual  processes." 

He  found  a  rapid  rise  in  the  rate  of  improvement  during  the 
early  practice,  an  observation  generally  confirmed  by  all  ex- 
perimenters. At  another  time  he  used  19  adult  subjects,  8 
men  and  11  women  students  in  Columbia  University,  giving 
them  each  day  for  seven  days  48  columns  of  ten  figures  each 
to  add,  in  all  2,592  additions.  These  subjects  were  able  on 
the  second  test  to  reduce  the  time  31%  and  the  errors  29%, 
with  a  total  improvement  of  29%.  Only  fifty-three  minutes' 
practice  was  actually  given.    Thorndike  says  of  this  test: 

"  That  the  practice  represented  by  only  2,592  additions  made  by 
an  educated  adult  whose  addition  associations  have  been  long  estab- 
lished and  often  used  should  produce  an  improvement  of  three- 
tenths  bears  witness  to  the  continued  plasticity  and  educability  of 
the  synapses  involved." 


CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC     175 

In  another  connection  he  has  stated  (127,  290)  : 

"  So  apparently  simple  an  ability  as  ordinary  addition  of  integers 
can  be  shown  to  require  analysis  into  at  least  nine  separate  abilities, 
each  of  which  probably  requires  further  analysis,  in  one  case,  into 
perhaps  ninety  component  ability-atoms." 

Similarly,  Donovan  and  Thomdike  (128,  134)  used  29  fourth- 
grade  pupils  and  found  in  a  practice  series,  given  two  periods 
of  two  minutes  each  per  day  for  three  school  weeks,  or  a 
total  of  thirty  two-minute  periods,  the  average  rose  from 
294  to  4^/2  examples  per  minute.  Kirby  (65,  24)  studied 
experimentally  732  children  in  the  fourth  grade,  testing  them 
before  and  after  sixty  minutes  of  practice,  after  the  method 
just  mentioned.  He  found  the  average  score  changed  from 
31  columns  with  24  correct  per  minute  to  50  columns  with 
37  correct.  The  children,  therefore,  gained  61%  in  attempts 
and  54%  in  correct  additions,  and  maintained  the  same  rate 
in  their  accuracy  and  in  their  speed,  almost.  Hahn  (128, 
134)  has  obtained  similar  results.  Kirby  also  gave  a  series 
of  drills  for  fifty  minutes  to  606  third  and  fourth  grade 
children.  At  the  beginning  the  children  averaged  40  simple 
divisions  per  minute,  with  37  correct,  an  accuracy  of  almost 
93%.  At  the  close  of  the  drill  series  they  performed  73 
divisions  per  minute  with  70  correct,  an  accuracy  of  almost 
97%.  They  had  gained  83%  in  speed  and  almost  90%  in 
their  accurate  results.  At  McLean  Hospital,  Wells  found 
that  ten  nurses,  five  men  and  five  women,  in  oral  addition 
of  digits  printed  one  above  the  other,  in  five  weeks,  practic- 
ing five  minutes  per  day,  six  days  in  the  week, — a  total  of 
150  minutes  of  practice,  increased  their  speed  nearly  100% 
and  maintained  about  the  same  rate  of  gain  in  accurate  work. 
The  five  women  on  the  first  day  performed  1,115  additions 
in  five  minutes,  while  the  five  men  reached  1,120.  On  the 
thirtieth  day  the  women  aggregated  2,210,  and  the  men  2,178 
additions.  The  lowest  score  at  first  was  made  by  a  woman 
who  got  150,  and  at  the  end  she  reached  280.  The  highest 
at  first  was  by  a  man  who  got  290  and  on  the  thirtieth  day 
he  went  to  540.  It  is  observed  that  the  lowest  and  the  highest 
made  practically  the  same  rate  of  gain.  Whitley  (134,  129) 
tested  nine  subjects  in  mental  multiplication,  giving  a  prac- 
tice series  of  three  examples  per  day  for  twenty  days,  omitting 
Sundays.  These  subjects  averaged  2.8  minutes  practice  per 
day,  or  a  total  of  fifty-six  minutes.  The  results  show  more 
than  100%  gain  in  speed,  with  no  ill  effect  upon  accuracy. 
In  the  Dumfries  schools,  Jeffrey  (61,  392)  selected  9  boys 
and  9  girls,  and  placed  them  in  three  groups  according  to 
their  mental  ability,  as  disclosed  in  previous   school   work. 


176     CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC 

Each  group  consisted  of  three  boys  and  three  girls,  the  groups 
being  classified  as  bright,  average  and  dull.  They  were  all 
given  fifteen-minute  drills  for  five  consecutive  days  and  then 
were  tested  with  the  following  results:  The  9  boys  made 
19,717  additions  and  the  9  girls  made  18,304,  with  126  errors 
by  the  boys  and  134  errors  by  the  girls.  The  group  results 
were:  bright  boys  gained  from  first  to  second  day,  49.1%, 
girls,  32.7% ;  average  boys,  28.2%,  girls,  32.3% ;  dull  boys, 
25.7%  ;  girls,  44.6%.  From  first  to  fifth  day  the  gains  were: 
bright  boys,  90.9%,  girls,  58.9%  ;  average  boys,  62.9%,  girls, 
67.2%;  dull  boys,  47.4%,  girls,  91.1%. 

Short  and  Long  Periods  of  Drill. — Kirby  tried  the  effect 
of  dividing  the  total  practice  time  into  periods  of  different 
lengths.  He  used  1,338  children  (65,  63)  and  his  results 
have  considerable  pedagogic  value.  Using  100  as  a  standard 
of  comparison  his  data  may  be  expressed  as  follows :  Gains 
made  with  22-minute  practice  periods,  100;  15-minute  periods, 
121 ;  6-minute  periods,  loi ;  2-minute  periods,  146.5.  These 
were  in  addition.  The  results  in  division  were:  from  20- 
minute  periods  taken  as  a  basis,  100;  lo-minute  periods, 
1 10.5;  2-minute  periods,  177.  That  is,  the  short  periods  of 
practice  scattered  over  more  days  give  a  higher  rate  of  gain. 
These  are  subject,  however,  to  discount  since  the  children 
during  the  longer  time  elapsing  from  the  first  to  the  last 
test  would  gain  more  from  their  regular  work  than  those 
taking  the  longer  single  periods  of  practice.  In  the  Whitley 
tests  already  referred  to  it  should  be  noted  that  after  twenty 
days,  with  practice  upon  only  three  examples  each  day,  the 
subjects  were  able  to  reduce  the  time  from  338  seconds,  with 
1.7  errors  per  example  to  135  seconds  with  1.4  errors  per 
example.  Thorndike  drilled  sixteen  subjects  continuously  on 
sixty  examples,  the  number  used  by  Whitley,  but  employing 
only  one  period  of  practice,  varying  from  2  to  12  hours.  His 
adult  subjects  took  an  average  of  352  seconds  per  example 
with  1.2  errors  on  the  first  test,  and  160  seconds  with  0.8 
errors  on  their  final  test.  Their  total  time  of  practice  averaged 
higher  than  did  those  tested  by  Whitley,  while  their  gain  was 
not  so  marked.  However,  it  is  to  be  noted  that  Thorndike's 
subjects  took  their  final  test  at  the  close  of  three  or  four 
hours  of  unrelenting  practice,  and  some  allowance  is  to  be 
made  for  their  jaded  condition. 

Permanency  of  Improvement  through  Drill. — Wells  tested 
for  the  permanence  of  improvement  in  the  adult  nurses  who 
had  taken  the  drill  from  January  to  April,  1910,  giving  six 
of  them  two  tests  in  December,  191 2,  after  a  lapse  of  2  years 
and    8    months    (128,  323).     In    January,   1910,  they    had 


CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC     177 

averaged  234  additions  in  their  first  test  and  274  on  the 
second,  while  their  average  score  in  the  final  test  in  April, 
1910,  had  been  447.  In  December,  1912,  these  same  adults 
on  their  first  test  scored  343  additions  and  the  next  day- 
raised  it  to  375.  Kirby  (65,  7  iff)  tested  258  of  his  fourth- 
grade  children  from  three  to  twelve  weeks  after  they  had 
relinquished  their  practice  and  found  them  able  to  do  as  well 
as  they  had  done  on  their  previous  final  tests.  He  again 
submitted  152  of  them  to  tests  in  September,  at  the  opening 
of  school  after  the  summer  vacation,  and  they  showed  a 
decided  loss  in  speed  with  some  decrease  in  accuracy.     Prac- 


TABLE  A 

Twenty  Leading  "Type-Errors,"  Made  by  Two  Hundred  and  Thirty- 
eight  Children  in  Addition  of  Digits,  Arranged  in 
Order  of  Frequency.    Compiled  from  Phelps  (92) 


Rank 

Combination 

Total 
number 

of 
errors 

Percent- 
age 
of  the 
attempts 

Number 

of 

children 

making 

the 

error 

Percentage 

of  the 

children 

making 

the 

error 

Number 

of 
children 
making 
the  error 

more 

than 

once 

Percentage 
of  children 
making  the 
error  two 
or  more 
times 

1 

9  plus  7 

395 

3.32 

120 

50.42 

63 

26.47 

2 

8  plus  5 

369 

3.10 

109 

45.80 

59 

24.79 

3 

9  plus  6 

320 

2.69 

94 

39.49 

49 

20.59 

4 

9  plus  3 

304 

2.55 

91 

38.23 

47 

19.75 

5 

9  plus  5 

298 

2.50 

86 

36.13 

38 

15.96 

6 

9  plus  8 

145 

2.44 

79 

33.19 

37 

15.55 

7 

8  plus  4 

142 

2.38 

79 

33.19 

33 

13.86 

8 

8  plus  7 

137 

2.30 

74 

31.09 

32 

13.44 

9 

6  plus  5 

135 

2.27 

72 

30.25 

31 

13.02 

10 

7  plus  5 

134 

2.25 

71 

29.83 

31 

13.02 

11 

7  plus  3 

120 

2.02 

67 

28.15 

26 

10.92 

12 

8  plus  8 

118 

1.98 

67 

28.15 

25 

10.50 

13 

7  plus  4 

232 

1.95 

60 

25.21 

25 

10.50 

14 

8  plus  3 

231 

1.94 

59 

24.75 

24 

10.08 

15 

7  plus  6 

186 

1.56 

58 

24.37 

23 

9.66 

16 

3  plus  3 

87 

1.46 

52 

21.85 

21 

8.82 

17 

7  plus  2 

158 

1.33 

52 

21.85 

21 

8.82 

18 

5  plus  3 

78 

1.31 

48 

20.17 

20 

8.40 

19 

6  plus  3 

71 

1.19 

48 

20.17 

17 

7.14 

20 

9  plus  2 

71 

1.19 

44 

18.48 

15 

6.30 

178     CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC 

tice  drills  of  fifteen  to  forty-five  minutes  were  sufficient  to 
restore  the  speed  of  the  former  tests.  Brown  (13)  tested 
his  practice  school  class,  already  discussed,  after  the  summer 
vacation  of  twelve  weeks  and  found  that  the  drilled  section 
in  September  was  able  to  raise  the  averages  made  in  June, 
while  the  undrilled  section  either  showed  no  gain  or  had 
retrograded. 

Type-Errors. — Phelps  (92)  reviews  the  Otis-Davidson  tests 
upon  270  children  in  the  eighth  grade  and  uses  their  data  in 
the  study  of  errors.  These  tests  were  given  in  the  grammar 
school  at  San  Jose,  Cal.,  and  238  sets  of  papers,  5,950 
separate  tests,  were  obtained.  In  Table  A  I  have  arranged 
the  chief  mistakes  in  addition  in  the  order  of  their  frequency. 

There  is  a  remarkably  consistent  showing  in  the  table, — 
the  combinations,  which  were  the  more  difficult  as  shown 
by  the  number  of  times  they  occur,  are  also  those  which 
are  made  by  the  largest  number  of  children  and  are  more 
often  repeated  by  the  same  child  than  the  other  combinations. 
Of  the  eight  schools  reported,  those  showing  the  highest 
speed  had  the  lowest  percentage  of  accuracy. 

Phillips  (95,  245),  out  of  440  returns  made  to  him,  gives 
the  following  list  of  difficult  combinations  of  the  digits  in 
addition:  157  find  9  plus  7  the  hardest;  88,  7  plus  8;  34,  6 
plus  7;  42  find  7  alone  troublesome;  18,  9  only;  26  are 
bothered  in  using  3,  6,  and  8;  327  mention  7,  and  204  give 
9  in  the  list  of  digits  they  find  hard  to  handle  correctly. 

In  1905  in  Budapest,  Ranschburg  (97)  tested  153  children, 
to  whom  he  gave  65  tests,  20  each  in  Addition  and  Subtrac- 
tion, 15  in  Multiplication,  and  10  in  Division,  in  an  attempt 
to  determine  which  fundamental  operation  is  the  most  diffi- 
cult to  school  children.  If  we  consider  accuracy  alone,  his 
order,  placing  the  easiest  first,  is :  Addition,  Multiplication, 
Subtraction  and  Division;  but  if  they  are  arranged  according 
to  speed  they  are:  Multiplication,  Addition,  Division  and  Sub- 
traction. If  both  speed  and  accuracy  be  combined  the  order 
is:  Multiplication,  68.75%;  Addition,  60.25%;  Division, 
50.65%  ;  and  Subtraction,  46.82%. 

Manner  of  Adding. — Amett  (5,  327fT)  tested  eight  adults 
in  their  habits  of  adding,  using  15  columns  of  2y  figures 
each.  He  had  them  add  for  thirty  minutes,  rest  a  few  min- 
utes and  then  add  for  thirty  minutes  more.  Nearly  200  col- 
umns were  added.  It  was  found  that  some  of  the  subjects 
employed  straight  addition,  following  the  columns  and  making 
as  many  additions  as  there  were  intervals  between  figures, 
while  others  used  combinations  of  digits.     Out  of  a  possible 


CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC     179 

use  of  840  combinations  one  announced  840  results,  while 
another  gave  only  519  results  for  a  possible  810,  having  made 
171  combinations  of  two  figures  and  43  combinations  of  three 
figures  each. 

Cole  gave  three  tests  to  35  persons  selected  at  random  to 
determine  their  habits  of  adding  upward  and  downward. 
The  first  test  consisted  of  20  columns  of  40  figures  each,  the 
subjects  adding  the  odd  columns  upward  and  the  even  ones 
downward.  The  downward  adding  required  15.3%  more  time 
than  the  upward  adding,  but  there  were  fewer  errors  in  it, — 
54%  of  the  errors  being  made  in  the  upward  and  46%  in 
the  downward  adding.  The  second  test  consisted  of  10  col- 
umns, identical  for  the  upward  and  the  downward  adding, 
and  a  third  tested  the  reading  of  numbers  from  left  to  right 
and  from  right  to  left  in  a  horizontal  line.  It  was  found 
that  the  established  habit  of  adding  upward  gave  more  speed 
but  resulted  in  greater  liability  to  error.  In  the  reading  of 
numbers  the  average  time  from  left  to  right  was  34.4  seconds 
with  62  errors,  while  the  reading  to  the  left  averaged  37.1 
seconds  with  a  total  of  36  errors.  The  usual  reading  habit 
afforded  greater  speed  to  the  right,  but  the  additional  atten- 
tion that  was  demanded  in  the  reversed  reading  resulted  in 
a  higher  accuracy   (23,  83!?). 

Socialisation  of  Arithmetic. — Paine  reports  a  recent  experi- 
ment in  Boston  with  some  sixth-grade  children  who  were  slow 
and  indolent  in  Arithmetic,  but  not  mentally  defective.  They 
were  chronic  **  failers "  and  were  particularly  deficient  in 
Arithmetic.  A  "  grocery  store  "  was  fitted  out  for  them  with 
enough  of  the  real  supplies  to  make  the  experiment  more  than 
symbolic.  It  was  found  that  the  children  took  on  new  life 
not  only  in  Arithmetic  but  in  their  Language  work  as  well. 
The  results  in  Arithmetic  are  given  as  follows: 

1.  Increased  accuracy  and  speed  in  computations. 

2.  Confidence  was  established  in  independent  solution  of 
problems. 

3.  A  good  drill  was  afforded  in  making  up  original  prob- 
lems. 

4.Some  valuable  training  was  gained  in  business  methods. 

Dooley  (37)  also  reports  some  interesting  work  in  the  suc- 
cessful motivation  of  Arithmetic  in  the  Massachusetts  Indus- 
trial School. 

The  Wording  of  Problems. — The  effect  upon  results  occa- 
sioned by  a  change  in  the  wording  of  problems  is  reported 
by  Phillips  (95,  268)  in  the  case  of  224  teachers  or  those 
preparing  to  teach.     When  40  problems  involving  Gain  or 


180     CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC 

Loss,  expressed  in  common  fractions,  was  given  them  8i.6% 
of  them  solved  all  correctly  while  the  others  averaged  5  prob- 
lems missed.  Later  the  same  problems  were  given  to  212  of 
the  same  group,  with  a  change  only  in  the  wording  from 
fractions  to  per  cent  and  62%  solved  all  correctly,  while  the 
others  averaged  3  missed.  During  the  past  year  Courtis  (31, 
4)  turned  his  attention  to  the  question  of  the  wording  of 
problems.  He  was  able  to  construct  twenty-one  varieties  of 
problems,  based  upon  a  single  situation,  by  changing  the  form 
of  the  question  and  the  relative  position  of  the  phrases  em- 
ployed, and  his  tests  showed  that  one  of  these  problems, 
measured  by  the  errors  made  by  children  in  solving  them, 
was  nineteen  times  as  difficult  as  another.  That  the  mere 
rearrangement  of  the  words  and  phrases  in  problems  causes 
such  wide  difference  in  results  should  have  careful  consid- 
eration from  both  authors  of  Arithmetics  and  teachers  of 
that  subject. 

Correlation  of  Abilities  in  Arithmetic. — Lobsien  (73)  con- 
cludes upon  some  experimental  studies  of  arithmetical  abilities 
with  the  following: 

1.  There  is  no  correlation  between  ability  to  remember  num- 

ber images  visually  and  the  ability  to  write  numbers. 

2.  The  greater  the  ability  for  solving  problems  in  the  head, 

the  weaker  is  the  memory  of  numbers  gained  through  the 
eye. 

3.  The  highest  correlation  exists  between  acoustic  number- 

memory  and  ability  to  write  numbers. 

4.  The  good  head-reckoner  generally  performs  written  work 

well,  and  znce  versa. 

5.  Acoustic  memory  of  numbers  and  ability  to  perform  opera- 

tions well  have  a  smaller  correlation. 

Stone  (117,  43)  believes  that  ability  in  any  fundamental,  with 
the  exception  of  Addition,  implies  ability  in  an  equal  degree 
in  the  other  fundamentals,  nearly,  and  he  found  that  many 
factors  influence  individual  abilities.  In  the  extensive  tests 
in  the  New  York  City  schools,  Courtis  reports  (30,  79)  that 
speed  and  accuracy  have  no  correlation  and  (p.  84)  "  it  was 
found  that  a  child  with  good  reasoning  ability  did  not  make 
mistakes  in  the  abstract  work."  So  far  as  his  analysis  of  the 
results  goes,  accuracy  is  dependent  upon  reasoning  and  simple 
reasoning  is  directly  related  to  ability  in  abstract  work,  high 
scores  in  "test  6"  (a  simple  reasoning  test)  being  associated 
with  high  scores  in  "test  7"  (a  test  in  the  fundamentals). 

Up  to  a  certain  critical  point  there  appears  to  be  a  definite 
correlation  between  a  good  knowledge  of  the  tables  and  ability 
to  work  speedily  in  the  abstract  examples,  and  a  lower  corre- 


CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC     181 

lation  with  accuracy.  The  curves  given  by  Courtis  for  the 
13,629  boys  and  the  13,542  girls  agree  quite  closely.  The 
evidence  which  he  submits  warrants  the  deduction  that,  in 
general,  a  knowledge  of  the  tables  makes  for  speed  and 
accuracy  up  to  a  certain  point,  beyond  which  other  factors 
play  such  an  important  part  that  further  knowledge  is  of  no 
benefit  (92-96). 

Winch  (137)  tested  four  schools  in  1909  and  one  in  1910 
in  London,  and  reports  his  findings  on  this  question,  as  the 
writer  has  gathered  them  together,  as  follows: 

TABLE  B 
Correlation  Tests  in  London  Schools 


Tests 

Grade 

Number 

of 

pupils 

Average 
age 

Correlation 
between 
accuracy 

and 
reasoning 

Eflfect  of  practice 

1 

7  and  8 

32  girls 

13 

.68 

Improvement.      No   "transfer" 
was  observed 

2 

3 

43  girls 

10 

.79 

Accuracy  gained 

3 

4 

38  girls 

10.5 

.69 

21%  gain  in  accuracy 

4 

4 

35  boys 

10 

.85 

20%  gain  in  accuracy 

5 

4 

72  boys 

10.25 

.736 

40%  gain  in  accuracy 

From  his  series  of  tests  it  appears  that  he  found  accuracy  in 
computation  to  accompany  good  reasoning  ability,  but  im- 
provement in  computation  did  not  affect  perceptibly  accuracy 
in  reasoning.  Starch  (114,  310)  gave  special  attention  to 
the  question  of  transfer  of  training  in  his  investigations,  and 
concludes : 

"  The  improvement  in  the  end  was  due  to  the  identical  elements 
acquired  in  the  training  series  and  directly  utilized  in  the  other 
arithmetical  operations." 

Comparison  of  Adults  and  Children. — Freeman  (40a) 
sought  in  some  experiments  with  14  adults  and  14  children, 
ranging  in  age  between  6  and  14  years,  to  determine,  if  pos- 
sible, how  children  differ  from  adults  in  the  elementary  scope 
of  attention,  and  also  what  differences  there  are  in  the  num- 
ber of  objects  which  may  be  grasped  in  a  single  act  of  atten- 
tion by  adults  and  children.  He  used  spots  of  light  thrown 
upon  a  screen  as  stimuli,  and  varied  the  time  exposure  from 
.018  to  .040  of  a  second  for  the  adults  to  more  than  a  second 
for  some  of  the  children.     He  found  the  range  of  attention 


182     CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC 

of  adults  and  children  to  be  nearer  together  than  is  generally 
supposed  (p.  309).  In  adults  he  thinks  the  range  may  aver- 
age 6,  although  his  observers  varied  from  4  to  7,  while  chil- 
dren between  12  and  14  years  of  age  will  average  5  and 
younger  children,  4  (p.  309).  He  gave  1,806  exposures  to 
his  adults,  while  Nanu  gave  only  100,  and  he  also  figured  out 
very  carefully  the  optimum  distance  at  which  to  place  his 
observers  from  the  screen.  For  these  reasons  he  thinks  his 
results  are  better.  Nanu  found  with  her  observers  a  decided 
tendency  to  underestimate  the  number  of  spots  shown;  only 
I  of  her  5  observers  overestimated  the  number,  while  11  of 
the  14  sitting  for  Freeman  overestimated  them.  He  reports 
one  pure  analytic  type,  four  mixed  types  with  strong  analytic 
inclination,  two  mixed  with  inclination  to  the  synthetic  type, 
and  seven  of  the  pure  mixed  variety.  He  differs  strikingly 
from  the  findings  of  Nanu,  who  reported  that  she  found  the 
synthetic  thinkers  always  inclined  to  underestimate  the  num- 
bers and  the  analytic  type  to  overestimate  them.  Of  Free- 
man's observers,  the  three  who  underestimated  the  numbers 
showed  no  tendency  toward  synthetic  thinking,  and  the  two 
who  were  at  all  inclined  to  synthesis  overestimated  the  spots, 
— one  in  94%  of  the  erroneous  judgments,  and  the  other  in 
86%  of  the  cases, — while  four  of  the  five  who  gave  evidence 
of  analytic  thought  underestimated  them. 

Some  of  the  introspections  seemed  to  show  that  the  ob- 
servers could  image  the  groups  of  objects  and  describe  them 
without  having  a  grasp  of  the  correct  number,  and  he  con- 
cludes that  the  number  name  is  not  essential  to  a  compre- 
hension of  a  group  of  objects.  "  Neither  the  word  nor  the 
name  is  necessary  for  the  number-concept." 

In  his  experiments  with  children  he  found  less  satisfactory 
results,  as  they  were  unable  to  give  reliable  introspections  and 
only  a  small  number  of  children  of  any  age  was  used.  Of 
the  14,  two  were  6  years  of  age;  two  were  7;  two  were  8; 
two  were  10;  four,  12;  one,  13;  and  one,  14.  As  these 
were  scattered  through  six  of  the  eight  grades,  with  no  repre- 
sentative in  the  fourth  and  sixth  grades,  it  gives  small  results 
for  each  grade.  These  children  had  been  taught  numbers 
upon  the  Russian  reckoning-machine,  so  they  were  not  en- 
tirely in  new  experiences.  He  found  it  necessary  to  exclude 
from  his  final  results  the  four  younger  children's  reactions, 
as  they  were  unreliable.  Of  the  remaining  10,  he  made  two 
groups  of  5  each  according  to  ages.  The  5  children  between 
8  and  10  years  of  age  showed  marked  differences  from  the 
5  who  were  12  to  14  years  old,  while  the  older  group  resem- 
bled very  much  the  adults,  both  in  range  of  attention  and 
in  its  behavior. 


CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC     183 

As  a  general  result  of  his  experiments  Freeman  concludes 
that: 

(i)  Children  as  a  rule  comprehend  a  number  of  objects  less 
correctly  than  adults. 

(2)  Children  prefer  a  horizontal  arrangement  of  objects  to 

be  seen. 

(3)  Children  show  a  more  rapid  decline  in  correct  answers 

as  the  number  of  objects  is  increased. 

(4)  The  range  of  attention  in  children  is  from  i  to  2  less 

than  in  adults. 

(5)  Children  underestimate  more  frequently  than  adults. 

(6)  Definite  arrangements   in  groups   is   less   favorable   for 

children  than  for  adults. 

(7)  Attention  in  young  children  is  very  irregular. 

(8)  No  correlation  was  found  between  school-talents  and  cor- 

rect answers. 

(9)  Groups  of  5  were  better  for  the  children  and  groups  of 

4  for  adults.  This  last  finding  agrees  with  the  conten- 
tion of  Lay  that  the  quadrate  form  is  the  best;  but 
Freeman  finds  it  only  true  for  adults,  rather  than  for 
school-children. 

It  must  be  confessed  that  we  do  not  yet  have  adequate  results 
to  justify  any  conclusions  upon  the  perception  of  simultane- 
ously presented  objects  as  a  basis  for  early  number  training. 
Too  few  observers  have  been  used  so  far. 

From  such  experiments  as  those  of  Freeman,  one  may  be 
led  to  infer  that  the  basis  for  the  difficulties  encountered  by 
children  and  adults  in  mastering  the  multiplication  table  lies 
in  the  inability  to  handle  numbers  in  groups  of  more  than 
5  or  6  readily.  To  master  the  table  of  7's  or  8's  or  9's,  one 
has  to  group  the  numbers  in  bundles  of  7,  or  8  or  9.  The 
pupil  usually  finds  the  numbers  below  6  rather  easy  in  com- 
parison with  numbers  between  6  and  10.  To  say  the  9's, 
one  has  to  group  the  numbers  up  to  90  in  bunches  of  9's, 
and  the  attention  has  to  pass  rapidly  over  the  groups  if  the 
learner  is  at  all  visually  minded. 

Individual  Differences. — There  is  to  be  seen  from  the  vari- 
ous experiments  reported  a  very  wide  divergence  in  number 
ability  among  school  children.  These  differences  are  shown 
in  the  aversion  of  some  to  the  subject  as  a  school  topic,  while 
others  choose  it  as  their  favorite  branch.  This  is  often  caused 
by  wrong  motivation  at  some  previous  time,  or  to  attitudes 
of  parents  toward  the  subject,  or  to  poor  teaching  in  a  lower 
grade,  in  the  case  of  those  who  dislike  it;  and  to  home  en- 
couragement, proper  motivation,  or  good  teaching,  or  per- 


184     CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC 

haps,  to  all  these  influences  combined  in  the  case  of  those 
who  prefer  it.  Frequently,  it  has  been  found  that  differences 
in  habits  of  thinking  cause  variations  in  school  interest  in 
Arithmetic.  No  doubt,  the  varying  degrees  of  interest  may 
sometimes  lie  far  back  of  school  experience,  in  the  child's 
opportunities  to  satisfy  his  inclinations  for  number  in  the 
nursery.  Provisions  that  are  made  for  the  natural  growth  in 
number  in  young  children  in  various  materials  afforded  in 
the  home,  coupled  with  an  active  interest  on  the  part  of  the 
mother,  often  determines  the  future  bent  of  the  child.  While 
no  other  subject  has  been  as  much  taught  in  the  modem 
school,  it  is  equally  true  that  in  no  other  subject  has  there 
been  so  much  bad  teaching.  So  to-day,  partly  through  im- 
perfect teaching  and  a  variety  of  pre-school  inclinations,  we 
find  in  the  school  grades  children  of  almost  every  degree  of 
advancement  in  the  same  grade.  Some  in  the  first  or  second 
grades  have  as  good  ability  in  numbers  as  others  in  the  sixth, 
seventh,  or  eighth  grades. 

Sex  Differences. — In  addition  to  the  effect  of  puberty  al- 
ready cited  from  Voigt,  a  number  of  observations  have  been 
recorded  upon  the  differences  in  arithmetical  abilities  depend- 
ent upon  sex.  Ballard  (9,  18)  says  that  in  a  series  of  tests 
in  the  London  schools  the  girls  showed  better  mechanical 
skill  in  the  solutions,  but  the  boys  did  the  problems  better, 
and  **  on  the  whole  the  boys  were  considerably  ahead  of  the 
girls."  Phillips  (93,  163)  tested  69  pupils  in  the  Granite 
Falls,  Minn.,  schools  from  March  to  May,  191 2,  and  found 
in  the  progress  made  in  drill  work  that  **  the  girls  did  better 
than  the  boy«  in  tests  in  fundamentals  "  and  the  boys  "  did 
better  work  in  reasoning,"  while  the  boys  made  a  greater  gain 
between  the  tests  than  did  the  girls,  their  gain  being  about 
24%  over  that  of  the  girls.  In  his  extensive  tests  in  New  York 
City,  Courtis  (30,  136)  found  also  that  "  the  girls  exceeded 
the  boys  in  the  speed  tests  in  multiplication,  but  they  fell  below 
them  in  accuracy  in  reasoning."  From  a  comparison  of  all 
the  scores  made  by  the  boys  and  the  girls  it  is  seen  that  the 
girls  excel  the  boys  in  mechanical  work  in  the  fundamentals 
and  the  boys  excel  in  simple  reasoning.  He  concludes :  "  Dif- 
ferences between  the  abilities  there  undoubtedly  are,  but 
whether  due  to  sex  or  to  environmental  influences  the  differ- 
ences are  too  slight  to  be  of  any  significance  so  far  as  present 
knowledge  goes."  Smith  (109)  reports  on  3,869  students 
tested  in  the  Normal  School  at  Cortlandt,  N.  Y.,  as  follows: 

Of  1,265  ^^^  put  on  examination,  58.7%  passed  with  an 
average  of  84.1%. 


CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC     185 

Of  2,603  women  put  on  examination,  50.6%  passed  with 
an  average  of  83.4%. 

He  also  gives  the  records  of  1,462  men  and  1,564  women  at 
Ypsilanti,  Mich.,  which  show  practically  no  distinctions  are 
to  be  made  in  Arithmetic  on  account  of  sex. 

The  Rice  Tests. — In  the  autumn  of  1902,  Dr.  J.  M.  Rice 
gave  a  test  of  eight  problems  to  6,000  school  children  in  seven 
cities  (98,  100).  The  children  were  chosen  from  grades 
four  to  eight  inclusive.  In  all,  18  schools,  some  in  the  slum 
districts,  some  in  the  better  districts  and  some  in  aristocratic 
neighborhoods,  were  included.  He  studied  the  effect  of  home 
environment,  size  of  classes,  total  time  per  day  given  to  Arith- 
metic, average  age  of  pupils,  forenoon  and  afternoon  periods 
of  recitation,  methods  of  instruction,  teaching  ability  of  the 
instructors,  and  concluded  that  none  of  these  was  the  deter- 
mining factor  in  securing  good  results  in  the  subject.  Rather 
surprisingly  he  puts  the  whole  responsibility  ultimately  upon 
the  supervision.  "  This  means  (p.  136)  in  other  words  that 
the  controlling  factor  in  the  accomplishment  of  results  is  to 
be  found  in  the  systems  of  examinations  employed,  some 
systems  leading  to  better  results  than  others."  He  found  wide 
variations  in  the  upper  grades,  mechanical  errors  increasing 
in  them,  with  a  decided  deterioration  in  the  5th  and  6th  grades. 
Cities  usually  ranked  with  their  individual  schools;  that  is, 
good  work  in  one  school  usually  signified  good  work  through- 
out the  whole  city. 

The  Stone  Tests.— Dr.  C.  W.  Stone  (117)  gives  detailed 
data  from  26  school  systems  scattered  well  over  the  United 
States  and  comprising  tests  in  Arithmetic  which  he  personally 
gave  to  152  classes  of  pupils  in  the  6A  grade.  Of  the  sys- 
tems tested,  6  were  located  in  New  England,  1 1  in  the  Middle 
East,  and  9  in  the  Middle  West.  The  tests  were  given  under 
controlled  conditions,  and  covered  the  fundamental  opera- 
tions and  the  children's  ability  to  reason  upon  the  solutions. 
With  6,000  sets  of  papers  to  study  he  draws  the  following 
conclusions : 

1.  The  net  result  of  arithmetic  work  in  the  first  six  years 

is  several  products,  rather  than  a  product.  The^  study 
called  Arithmetic  makes  demand  upon  a  plurality  of 
abilities  (p.  43). 

2.  There  is  a   great  variability  in  the  products  of  different 

systems,  and  greater  still  among  individuals  in  any  sys- 
tem. The  variability  among  boys  does  not  appreciably 
differ  from  that  among  girls. 


186     CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC 

3.  The  possession  of  a  certain  amount  of  ability  by  a  system 
is  a  better  guarantee  of  the  same  amount  of  another 
ability  than  the  possession  of  a  certain  amount  of  ability 
by  an  individual  is  that  he  will  have  the  same  amount 
of  another  ability. 

He  agrees  with  Rice  upon  the  eifect  of  good  supervision, 
but  found  different  results  on  the  tests  in  reasoning  (p.  45). 
He  had  the  courses  in  Arithmetic  in  these  26  systems  rated 
for  him  by  21  professors  and  graduate  students  in  education 
(p.  71)  and  compares  the  results  of  his  tests  with  these 
ratings.     He  says: 

"The  situation  seems  to  be  that  the  course  of  study  is  not  at 
present  the  factor  that  it  ought  to  be  in  producing  abilities.  In 
certain  systems  it  is  evidently  working  well,  but  in  others  there  is 
a  wide-spread  disparagement  between  ex'cellence  in  abilities  and 
excellence  in  the  course  of  study."  "The  course  of  study  may  be 
the  most  important  single  factor  but  it  does  not  produce  abilities 
unless  taught.  The  other  essential  features  for  successful  teaching 
are  teachers  and  children  of  usual  abilities,  a  reasonable  time  allot- 
ment, intelligent  supervision  and  adequate  measurement  of  results 
by  tests   (p.  91). 

The  Courtis  Standard  Tests. — Following  immediately  upon 
the  results  obtained  by  Stone,  Dr.  S.  A.  Courtis  began  by 
giving  the  Stone  tests  to  317  girls  in  the  Liggett  school  at 
Detroit,  in  1908  and  1909.  These  girls  were  scattered  through 
the  grades  from  the  3rd  to  the  13th.  Using  his  results  as 
a  basis,  in  September,  1909,  he  devised  a  new  set  of  tests 
covering  speed  in  each  of  the  four  fundamental  operations, 
one  in  copying  figures,  two  in  reasoning  and  one  general 
test  in  all  four  fundamentals.  Under  controlled  conditions 
these  were  given  to  the  same  school  in  September,  1909, 
and  in  June,  1910.  Among  other  results  he  found  as  did  Rice 
that  the  6th  grade  was  a  "  notoriously  difficult "  one,  although 
it  had  ranked  high  as  a  5th  grade  the  previous  year  (29,  p. 

361). 

From  this  beginning,  Courtis  came  to  believe  that  a  uniform 
standard  test  could  be  devised,  so  he  sought  during  the  school 
year,  1910-11,  to  establish  such  a  standard.  He  gathered 
papers  from  near  9,000  children  in  from  60  to  70  schools 
scattered  in  10  states.  These  children  had  been  given  his  8 
tests  and  a  "  standard "  table  was  constructed  from  the 
results,  which  is  given  here  after  corrections  have  been  made 
in  it  from  later  facts  gathered  altogether  from  66,837  children 
and  revised  to  August,  1913. 

From  Table  C  it  will  be  seen  that  a  6th  grade  child 
should  be  able  to  give  correctly  50  combinations  in  addition 
of  digits,  38  in  subtraction,  37  in  multiplication  and  division, 


CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC     187 

and  copy  figures  at  the  rate  of  92  per  minute.  The  scores 
from  grades  3  to  8  inclusive  are  revised  to  suit  the  figures 
from  66,837  tests,  while  the  others,  I  understand,  are  those 
made  out  from  the  first  studies  gathered  from  19  cities. 


TABLE  c 


Courtis  Standard  Scores,  Showing  What  a 
Should  be  Able  to  Accomplish  in 

Child  in  Each  Grade 
One  Minute 

Grade 

Number  of 

simple 
additions 

Number  of 

simple 
subtractions 

Number  of 
simple 
multiplications 

Number  of 

simple 

divisions 

Number  of 
figures 
copied 

1 

(6) 

(6) 



(29) 

2 

(21) 

(12) 

(10) 

(12) 

(51) 

3 

26 

19 

16 

16 

63 

4 

34 

25 

23 

23 

75 

5 

42 

31 

30 

30 

84 

6 

"  50 

38 

37 

37 

92 

7 

58 

44 

41 

44 

100 

8 

63 

49 

45 

49 

108 

9 

(65) 

(50) 

(50) 

(50) 

(120) 

10 

(57) 

(45) 

(43) 

(46) 

(112) 

11 

(59) 

(47) 

(44) 

(48) 

(114) 

12 

(61) 

(48) 

(44) 

(49) 

(112) 

13 

(71) 

(56) 

(50) 

(56) 

(116) 

14 

(74) 

(51) 

(58) 

(59) 

(124) 

The  New  York  City  Tests. — From  March  15  to  April  26, 
1912,  Courtis  applied  his  tests  in  New  York  City  to  a  list 
of  33,350  pupils,  representing  a  school  register  of  40,000 
pupils,  or  about  one-tenth  of  the  city  school  population  from 
the  4th  up  to  the  8th  grade.  Representative  schools  were 
selected  in  various  parts  of  the  city;  21  schools  furnishing 
380  classes  with  12,147  pupils,  were  in  Manhattan  borough; 
9  schools,  with  148  classes  and  4,488  pupils,  were  in  the 
Bronx;  18  schools,  with  315  classes  and  10,243  pupils,  were 
in  Brooklyn ;  2  schools  with  2^  classes  and  646  pupils  were 
in  Richmond  borough ;  and  2  schools  with  37  classes  and 
1,145  pupils  in  Queens.  From  these,  28,669  complete  returns 
were  received,  but  only  27,171  were  tabulated  because  of 
apparent  irregularities  in  the  remainder.  Courtis  went  to 
New  York  and  personally  took  charge  of  the  tests  at  the 
request  of  the  "  Hanus  Committee  in  Charge  of  Educational 


188     CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC 

Aspects  of  School  Inquiry."     His  purpose  was  to  determine, 
if  possible,  the  following: 

1.  The  standard  of  achievement  in   fundamental  operations 

with  whole  numbers,  and  in  simple  reasoning. 

2.  The  relative  achievement  of  the  schools  tested,  as  measured 

by  standards. 

3.  The   relative   achievement  of  grades   and   individuals,   as 

measured  by  standards,  so  far  as  is  necessary  to  indicate 
to  teachers,  principals,  and  superintendents  how  such 
knowledge  could  be  used  to  make  their  work  more 
efficient. 

4.  The  relative  achievement  of  New  York  City  schools  as  a 

whole,  as  measured  by  standards  derived  from  tests  in 
other  cities. 

In  the  following  table  will  be  given  the  score  of  the  various 
grades  in  each  one  of  the  eight  tests  used  and  at  the  same 
time  the  highest  score  made  by  any  pupil,  with  some  other 
data  that  may  assist  in  one's  understanding  the  gross  results 
of  the  experimental  study.  It  is  not  possible  for  us  to  enter 
upon  any  comprehensive  consideration  of  the  minor  details 
reported  by  Courtis  in  his  158-page  booklet.  A  few  findings 
have  already  been  mentioned  and  a  few  others  remain  to  be 
noted  after  the  table  is  studied. 


TABLE  D 
Correct  Score  Averages  Made  by  the  Different  Grades  in  the 


New 

York  City  Schools 

Grade 

4th 

5th 

6th 

7th 

8th 

9th 

10th 

nth 

12th 

Numb 

5396 

5386 

5670 

4771 

4502 

440 

257 

179 

120 

Avera 

ge  age  of  the  pupils  . . 

10.5 

11.3 

12.4 

13.9 

14.6 

No. 

Kind  of  test 

Max. 
Score 

1 

Addition,— speed... 

125 

41.9 

50.2 

56.9 

62.2 

69.5 

71.6 

71.7 

73.9 

74.2 

2 

Subtraction, — speed 

125 

29.5 

36.8 

41 

45.8 

52.2 

52.2 

55.3 

55.2 

54.1 

3 

Multiplication, — 

125 

28.7 

35 

38.3 

40.9 

45.8 

46.5 

46.8 

48.6 

46.6 

4 

Division, — speed  . . , 

125 

26.6 

34.7 

39.7 

44.6 

50.9 

52.5 

52.7 

54.2 

55 

5 

Copying  figures, — 
speed 

205 

75.4 

85.5 

92.5 

100 

106.8 

98.8 

104.5 

109.4 

105  6 

6 

Reasoning, — 
one  step 

16 

1.8 

2.3 

3 

3.7 

4.4 

5.1 

5.1 

5.1 

5  7 

7 

Fundamentals 

19 

4.2 

5.8 

7 

8.5 

10.1 

10.9 

11.5 

10.5 

11 

8 

Reasoning, — 

two  steps 

8 

.^« 

1.3 

1.7 

2.1 

2.5 

2.6 

2.7 

2.7 

' 

CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC     189 

It  must  be  borne  in  mind  that  the  Courtis  tests  are  exactly 
the  same  for  all  the  grades  and  that  he  holds  that  a  score 
of  forty  answers  per  minute  means  double  the  ability  that 
gives  only  twenty  per  minute  (p.  17),  and  to  change  from 
20  to  25  answers  corresponds  to  a  change  from  40  to  45 
per  minute.  The  errors  that  were  shown  in  the  papers  he 
classifies  as  follows: 

1.  Carelessness  in  bringing  down  the  wrong  figure  in  division 

or  placing  partial  products  in  multiplication  under  the 
wrong  figure,  12.5%. 

2.  Copying  incorrectly,  reversing  figures,  as  639  for  693,  8%. 

3.  In  fundamental  combinations,  50%. 

4.  Scattered,  such  as  errors  in  carrying,  etc.,  all  the  rest. 

Many  errors  were  found  in  handling  zero.  It  is  worthy  of 
mention  here  that  in  Columbia  University,  in  the  summer  of 
1910,  41  adult  graduate  students,  teachers  and  superintendents 
were  tested  and  18  made  104  mistakes  in  zero  combinations 
particularly  when  zero  occurred  in  the  multiplier. 

He  found  in  the  New  York  schools,  as  has  been  found 
wherever  tests  have  been  applied,  that  the  work  in  funda- 
mentals is  very  low  both  in  speed  and  in  accuracy  in  com- 
putation and  simple  reasoning.  The  critical  period  for  the 
mastery  of  these  fundamentals  seems  to  lie  down  in  the  lower 
grades.  In  the  introduction  to  the  Courtis  Report,  Dr.  Hanus 
calls  attention  to  the  low  degree  of  efficiency  in  the  schools 
and  says :  "  Children  of  every  level  of  ability  are  found  in 
every  grade  and  differences  between  individuals  greatly  ex- 
ceed the  difference  between  grades."  For  this  reason  the 
fundamentals  should  be  adapted  to  the  individuals  and  they 
should  be  taught  in  the  light  of  individual  capabilities.  "  This 
condition  is  universal  and  is  not  due  to  lack  of  effort  or  other 
conditions  that  could  be  easily  removed,  but  to  a  neglect  of 
one  basic  factor, — the  difference  in  the  powers  and  capabilities 
of  children,"  (p.  ']6'),  because  children  are  already  at  school 
entrance  highly  specialized  in  their  mental  habits.  Courtis 
thinks  (p.  130)  that  a  more  simple  and  practical  course  in 
Arithmetic,  based  directly  upon  the  social  needs  of  the  chil- 
dren, would  influence  for  good  a  greater  number  of  both 
children  and  teachers. 

The  Boston  Tests. — The  same  eight  tests  were  given  by 
Courtis  to  29  of  the  schools  in  Boston,  in  October,  191 2, 
and  again  in  March,  1913.  More  than  500  classes  and  about 
25,000  children  were  included.  These  tests  were  designed  to 
determine  the  following  facts: 


190     CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC 

1.  The  standard  of  arithmetical  work  in  the  Boston  schools, 

and  their  comparative  standing  with  other  schools  already- 
tested. 

2.  The  nature  and  degree  of  change  produced  by  six  months 

regular  work  in  Arithmetic. 

3.  The   effect   of   certain   special   methods   of  individual  in- 

struction. 

Supt.  Dyer  devotes  only  thirteen  pages  of  his  recent  report 
(38)  to  these  tests,  yet  some  valuable  facts  are  disclosed: 

1.  Boston  is  lower  in  abstract  work  and  higher  in  reasoning 

than  New  York. 

2.  A  comparison  of  the  tests  in  March  with  those  six  months 

earlier  shows  that  53%  of  the  pupils  made  improvement, 
30%  stood  still  and  17%  lost. 

3.  Four  special  methods  were  employed  in  the  experimental 

study : 

(a)  In  one  group,  each  teacher  took  one  period  per  week 

to  work  with  individual  pupils  who  seemed  from 
the  October  tests  to  need  particular  strengthening 
upon  certain  points.  This  was  continued  for 
twelve  weeks,  and  the  group  as  a  whole  gained 

14%. 

(b)  A    second    group    had    the    assistance   of    an    able 

specialist,  one  to  each  school  being  assigned  to  the 
task  of  giving  individual  help  to  children  sent  by 
the  regular  teachers.  This  also  ran  for  twelve 
weeks,  and  2,187  children  out  of  3,443  received 
individual  assistance,  in  60,000  interviews  lasting 
15  minutes  each.  The  report  says  this  method  did 
not  yield  the  results  expected. 

(c)  A  third  group  pursued  regular  work  with  no  reme- 

dial help,  and  were  used  as  a  "  control "  group  in 
measuring  the  others. 

(d)  A  fourth  group  was  formed  in  March,  of  pupils  who 

had  not  taken  the  October  tests,  and  they  were 
given  special  daily  drills  on  the  fundamentals  for 
ten  minutes  each  day  during  the  regular  recitation 
period.  In  the  March  tests  this  group  attempted 
fewer  problems  and  examples  than  the  others  and 
"this  method  was  the  least  effective." 

A  comparison  of  the  three  drilled  groups  with  the  "  control " 
group  shows  that: 

I.  Group  one,  in  which  the  regular  teachers  devoted  one  period 
per  week  to  a  class  drill  upon  fundamentals,  exceeded  the 
"  control "  group  by  8%  in  accuracy. 


CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC     191 

2.  Group  two,  in  which  a  special  teacher  gave  individual  help, 

exceeded  it  by  5%. 

3.  Group   four,   in  which  the  ten-minute  drill  upon   funda- 

mentals preceded  the  test,  exceeded  it  by  only  3%. 

These  facts  seem  to  emphasize  the  value  of  systematic  work 
under  the  regular  teacher  and  to  discount  the  common  prac- 
tice of  "  cramming  "  for  tests  and  examinations. 

Personal  Observation  of  the  Recent  Boston  Tests. — On 
Thursday,  January  8,  1914,  the  writer  went  to  Boston  to 
observe  the  giving  of  the  new  Courtis  tests  which  were  to  be 
given  to  more  than  20,000  children,  beginning  that  day  under 
the  supervision  of  Miss  Rose  A.  Carrigan,  of  the  Boston 
Normal  School.  Three  schools  were  selected,  one  mixed  and 
one  each  in  which  the  boys  and  girls  are  taught  separately. 
Miss  Carrigan  had  furnished  me  a  list  of  all  the  schools  to 
be  tested  each  day,  so  I  went  on  Thursday  without  her 
knowledge  to  schools  of  my  own  selection.  In  all,  I  saw  the 
tests  given  to  8  rooms,  5th,  6th,  and  7th  grades,  350  children 
being  present  and  taking  the  tests.  I  saw  15  of  the  young 
lady  "  cadets  "  doing  the  testing  at  the  three  schools,  two 
working  together  in  each  room.  One  would  explain  the  sig- 
nals to  be  observed  while  her  assistant  manipulated  the  signal- 
box  and  announced  the  time  to  start  and  quit.  There  was 
splendid  management  in  handling  the  tests  and  I  have  no  criti- 
cism to  offer  on  the  fairness  of  the  application  of  the  tests. 
The  children  seemed  to  enjoy  the  game  and  so  far  as  I  could 
see  took  no  notice  of  my  presence  or  that  of  the  master  who 
accompanied  me  to  their  room.  53  cadets  were  busy  in  these 
tests  after  being  trained  by  Miss  Carrigan  for  three  days. 
Another  test  was  given  the  second  week  in  April,  1914, 
but  it  will  be  several  weeks  before  results  can  be  known. 
These  tests  are  to  (i)  set  definite  standards  for  each  grade, 
(2)  measure  the  results  of  each  teacher's  work,  (3)  assist 
in  motivating  the  children's  work,  and  (4)  to  furnish  some 
studies  in  the  grading. 

Criticism  of  Standardisation. — It  is  proper  that  I  give  some 
discussion  upon  the  general  proposition  to  attempt  a  "  stand- 
ardization "  of  the  achievements  in  Arithmetic,  which  is,  after 
all,  only  a  subordinate  notion  of  "  standardization "  in  all 
studies.  There  appears  a  strong  tendency  to-day  to  try  to 
standardize  everything,  products  and  producers  alike,  and 
educational  circles  seem  to  have  been  caught  in  the  current. 
The  trades  are  said  to  have  a  definite  number  of  doors  for 
a  carpenter  to  hang  in  a  day,  or  a  specific  product  to  be 
turned  out,  beyond  which  a  good  workman  will  not  endeavor 
to  go.     With  discriminating  accuracy  there  are  attempts  to 


192     CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC 

"  standardize  "  in  this  world-movement  not  only  Arithmetic, 
but  also  penmanship,  compositions,  cattle,  peanuts  and  parents ! 

Courtis  has  selected  a  certain  line  of  level  from  the  thous- 
ands who  have  been  tested  by  his  method  and  sets  these  levels 
up  as  a  standard  for  the  schools  everywhere.  This  would 
not  seem  so  serious  did  we  not  find  him  saying  (31,  14)  : 

"  It  should  be  noted  that  standards  will  not  produce  uniform 
products  unless  they  are  treated  both  as  goals  to  be  reached,  amd 
as  limits  not  to  be  exceeded." 

He  points  out  that  the  average  score,  for  instance,  of  11,059 
8th  grade  pupils  on  test  No.  7  was  9.5  examples  correctly 
done,  but  38%  of  these  pupils  had  been  so  overtrained  that 
they  exceeded  this  score  by  10%  to  100%.  "  These  high 
scores  of  school  children  represent  waste  effort."  "  When 
standard  ability  has  been  attained,  additional  degrees  of  me- 
chanical skill  are  products  of  the  least  importance." 

While  the  writer  is  in  complete  sympathy  with  every  at- 
tempt to  place  Pedagogy  upon  a  sound  and  genuine  scientific 
basis,  he  does  not  consent  willingly  to  the  effort  to  "  Pro- 
crusteanize  "  the  schools  by  requiring  them  to  be  measured 
and  directed  by  semi-arbitrary  standards.  Abilities  of  chil- 
dren are  too  widely  variant  and  future  callings  are  too  diverse 
for  us  to  agree  that  each  child  in  the  great  public  school 
system  shall  be  moulded  into  the  same  set  form  in  Arith- 
metic, or,  indeed,  in  any  school  subject,  by  accepting  stand- 
ards which  are  to  be  reached  but  not  exceeded.  The  present 
movement  is  awakening  much  interest  among  the  school 
public  and  must  result  in  good.  Its  chief  temptation  lies  in 
the  extravagant  application  of  some  of  its  obviously  useful 
features  until  the  practical  educator  will  be  led  to  reject  even 
what  help  it  should  be  able  to  offer. 

The  selection  of  a  semi-arbitrary  standard  in  Arithmetic 
has  gone  no  farther  than  the  four  fundamental  operations 
and  simple  problems  in  reasoning,  and  has  not  approached 
fractions,  denominate  numbers,  ratio,  percentage,  mensura- 
tion,— yet  to  me  it  seems  much  like  taking  the  average  per- 
acre  corn  crop  of  the  country,  or  more  correctly,  of  a  few 
sections,  and  setting  this  up  as  a  "  standard  "  to  be  reached 
but  not  exceeded  by  farmers  everywhere!  Boys'  Corn  Clubs 
have  shown  the  com  raisers  the  fallacy  of  taking  "  aver- 
ages "  as  standards.  It  is  a  little  similar  in  educational  work. 
As  land  differs  in  fertility,  so  children  differ  in  Begabungen  or 
talents.  As  farmers  differ  in  their  methods  of  cultivation, 
so  parents  and  teachers  differ  in  Erziehung.  We  may  have 
to  leave  the  Begabungen  with  the  biologist  and  the  eugenist; 
but  with  more  fertile  methods  of  instruction,  with  a  more- 
socialized  curriculum,   it  is  possible  for  these  experimental 


CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC     193 


studies  to  become  "  stepping-stones  "  upon  which  we  may  rise 
to  higher  things.  At  all  events  we  should  recognize  and 
respect,  nay  more,  develop  and  encourage  the  individuality 
in  every  child. 

TABLE  E 

Distribution  of  Arithmetical  Topics  in  the  Grades  in  American  Cities* 

All  figures  are  to  be  read  as  percentages 


Topic 

Cities 

laving 

it 

^ 

1 

CM 

i 

II 

.  1 

.c 

00 

^ujulj^rs  introduced       

all 

71.5 

22 

6.5 

all 

5 

78      1 

7       .. 

all 

14 

21 

17 

21      2 

7       .. 

all 

2 

10      6 

3      23 

2 

*  * 

T>prim5il  frartion^  taiipht             ... 

all 

9      3 

2      40 

13 

5 

all 

5 

8      11 

20 

S6 

Pprrpntapf    ppnpral  cases 

all 

5      3 

0      45 

20 

' ' 

^imnl«»  intprfst                   

all 

5      20 

45 

30 

64.7 

5 

40 

55 

Commission 

all 

.       15 

55 

30 

all 

.       10 

50 

40 

all 

40 

60 

42.8 

23 

77 

Stocks  and  bonds 

71.4 

-■-I, 

__ 

5 

23 
20 

77 

60.7 

75 

57.1 

28 

7? 

Simple  proportion 

all 

12 

21 

67 

42.8 

. 

24 

76 

Mensuration, — of  plane  figures 

all 

10      1 

2      20 

30 

28 

^Mensuration  — of  solids 

all 

..     12 

18 

25 

45 

Square  root  . .          

all 

15 

85 

28.5 

. 

25 

75 

all 

12      2 

\3      25 

18 

12 

Least  common  multiple 

all 

15     : 

\7      34 

10 

4 

Greatest  common  divisor 

all 

12    : 

J7      37 

10 

4 

35.7 

•• 

•• 

28 

72 

*  In  the  above  table  the  distribution  of  the  topics  is  given  for  28  American  cities 
reporting  to  the  American  Committee  of  the  International  Commission  on  the  Teach- 
ing of  Mathematics,  and  published  in  the  U.  S.  Bureau  of  Education  Bulletin  No.  13, 
1911,  pp.  16-65,  75-78.  These  cities  represent  about  one-tenth  of  the  scholastic  popu- 
lation in  the  United  States. 


194     CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC 


TABLE  F 
Important  Facts  About  the  Experimental  Studies  in  Arithmetic 


Investigator 

Country 

Subjects 

Date* 

Purpose  of  experiments 

1.  Phillips.  D.  E. 

America 

260  children 

1897 

Popularity  of  arithmetic 

2.  Lewis 

England 

8,000 

1913 

" 

3.  Messenger 

America 

6  adults 

1903 

Kinds  of  thinkers 

4.  Nanu 

Germany 

7        " 

1904 

.. 

5.  Freeman 

" 

14        " 
14  children 

1910 

Differences  in  adults  and  chil- 
dren 

6.  Lobsien 

" 

1913 

Sex  differences 

7  Ballard 

England 

1912 

4<                              <i 

8.  Smith 

America 

3.869  adults 

1895 

" 

9.        "     

" 

9,307        " 

1895 

" 

10.  Phillips,  F.  M. 

69  children 

1913 

" 

11.  Courtis 

" 

27,171 

1912 

" 

12.  Voigt 

Germany 

1912 

4,                                .< 

13.  Brown 

America 

51  children 

1911 

Effect  of  drill 

14.       "      

" 

222       " 

1912 

" 

15.  Phillips,  F.  M. 

" 

69        " 

1913 

" 

16.  Starch 

" 

15  adults 

1911 

•• 

17.  Thorndike 

" 

33        " 

1908 

" 

18. 

" 

19        " 

1910 

.< 

19.  Donovan  and 
Thorndike..  . 

•• 

29  children 

1913 

,. 

20.  Kirby 

" 

1.338       " 

1912 

" 

21.  Hahn 

" 

192        " 

1913 

" 

22.  Wells 

•• 

10  adults 

1910 

" 

23.  Whitley 

" 

9       " 

1912 

" 

24.  Jeffrey 

Scotland 

18  children 

1912 

" 

25.  Kirby 

America 

1.338        " 

1912 

Effect  of  long  and  short  drills 

26.  Whitley 

" 

9  adults 

1912 

" 

27.  Thorndike.... 

" 

16        " 

1913 

" 

28.  Wells 

" 

6        " 

1912 

Permanency  of  improvement 

29.  Kirby 

" 

258  children 

1912 

" 

30.  Brown 

" 

51 

1911 

«                        « 

31.  Ranschburg  . . 

Hungary 

153       " 

1909 

Relative  difficulty  of  fundamen- 
tals 

32.  Phillips,  D.  E. 

America 

440        " 

1897 

Relative  difficulty  of  fundamen- 
tals 

*  The  dates  given  are  those  for  the  year  in  which  the  experiments  were  made  when 
these  were  obtainable;   otherwise,  for  the  year  when  the  results  were  published. 


CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC      195 

TABLE  F — Continued 
Important  Facts  About  the  Experimental  Studies  in  Arithmetic 


Investigator 

Country 

Subjects 

Date* 

Purpose  of  experiments 

33.  Phelps 

« 

270        " 

1912 

Relative  difficulty  of  fundamen- 
tals 

34.  Cole 

35  adults 

1912 

Adding  upward  and  downward 

35.  Arnett 

8       " 

1905 

Manner  of  adding 

36.  Paine 

12  children 

1913 

Motivationthrough  store-keeping 

37.  Phillips,  D.  E. 

" 

224 

1897 

Effect  of  change  in  wording 

38.  Courtis 

" 

317        " 

1909 

" 

39.  Winch 

England 

32  girls 

1909 

Correlation  and  transfer  of  abil- 
ities 

40.       "      

" 

43        " 

1909 

Correlation  and  transfer  of  abil- 
ities 

41.       "      

" 

38        " 

1909 

Correlation  and  transfer  of  abil- 
Ues 

42.       "      

" 

35  boys 

1910 

Correlation  and  transfer  of  abil- 
ties 

43.       "      

" 

72        " 

1910 

Correlation  and  transfer  of  abil- 
ties 

44.  Rice 

America 

6.000  children 

1902 

Comparative  results  of  schools 

45.  Stone 

" 

6,000       " 

1908 

Results  of  six  years'  work 

46.  Courtis 

" 

317       " 

1909 

Speed,  accuracy  and  reasoning 

47.       "      

" 

9.000        " 

1911 

Standardization  of  aims 

48.        "       

" 

27.171 

1912 

Abilities  as  related  to  standards 

49.        "       

" 

25,000        " 

1913 

" 

50.  Carrigan 

" 

20.646        " 

1914 

«                     i<               « 

*  The  dates  given  are  those  for  the  year  in  which  the  experiments  were  made  when 
these  were  obtainable;    otherwise,  for  the  year  when  the  results  were  published. 


V.  Discussion  and  Pedagogical  Deductions 

Several  facts  seem  to  stand  out  quite  clearly  in  these  experi- 
mental studies.  Some  of  the  more  prominent  ones  are:  (i), 
Inadequate  results  are  now  being  obtained  in  Arithmetic  in 
practically  all  schools;  (2),  Courses  of  study,  judged  by 
experts  as  superior,  give  no  better  results  than  those  judged 
as  inferior;  (3),  The  better  educated  teachers  are  not  having 
more  success  than  the  less  educated  ones;  (4),  Small  ex- 
penditure of  time  devoted  to  drill  upon  fundamentals  under 
optimal  conditions  gives  surprising  mechanical  skill,  which 
remains  fairly  permanent;  (5),  The  paradoxical  situation  has 
received  no  adequate  and  satisfactory  explanation  up  to  the 
present  time. 


196     CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC 

With  painstaking  care,  Rice,  in  his  pioneer  investigations 
a  decade  ago,  eliminated  one  by  one  the  elements  that  he 
thought  had  any  bearing  upon  the  cause  of  such  poor  mastery 
of  Arithmetic  and  finally  left  the  burden  upon  the  superin- 
tendent and  the  system  of  examinations  employed  in  the 
schools.  However,  I  believe  he  was  far  from  the  proper 
diagnosis.  The  whole  trouble,  when  we  come  to  the  ultimate 
analysis  of  the  case,  lies  in  the  failure  of  the  pupils  to  com- 
prehend what  they  pass  over  in  Arithmetic, — they  lack  an 
orderly  association  of  the  principles  of  the  subject,  and  they 
do  not  have  their  fundamental  and  hierarchical  systems  of 
mathematical  habits  well  established.  Bryan  and  Harter  (i6a, 
360)  tell  us  that  "A  man  is  organized  in  spots — or  rather  in 
some  spots — far  more  than  in  others.  This  is  true  structurally 
and  functionally."  "  Some  habits  are  knit  together  in  a 
hierarchy." 

"There  are  a  certain  number  of  habits  which  are  elementary- 
constituents  of  all  the  other  habits  within  the  hierarchy.  There  are 
habits  of  a  higher  order  which,  embracing  the  lower  as  elements, 
are  themselves  in  turn  elements  of  higher  habits.  A  habit  of  any 
order,  when  thoroughly  acquired,  has  physiological  and,  if  conscious, 
psychological  unity.  The  habits^  of  lower  order  which  are  its  ele- 
ments tend  to  lose  themselves  in  it,  and  it  tends  to  lose  itself  in 
habits  of  higher  order  when  it  appears  as  an  element  therein." 

Proficiency  in  Arithmetic  rests  pre-eminently  upon  a  mas- 
tery of  habits  associated  in  hierarchical  fashion.  Here,  per- 
haps more  than  in  any  other  school  subject,  does  a  ready 
habitual  response  indicate  efficiency,  and  as  Horace  Mann 
might  say  (52a,  236):  "One  former  is  worth  a  hundred 
reformers."  With  our  new  psychology  we  are  needing  a  few 
new  canons  for  Arithmetic.    Bryan  and  Harter  say  (i6a,  375)  : 

"  Now    the    ability    to    take    league    steps     .      .  ,    in    addition, 

.  ,  plainly  depends  upon  the  acquisition  of  league-stepping 
habits.  No  possible  proficiency  and  rapidity  in  elementary  processes 
will  serve.  The  learner  must  come  to  do  with  one  stroke  of  atten- 
tion what  now  requires  half  a  dozen,  and  presently  in  one  still 
more  inclusive  stroke,  what  now  requires  thirty-six.  He  must 
.  .  acquire  a  system  of  habits  corresponding  to  the  system 
of  tasks.  When  he  has  done  this  he  is  master  of  the  situation  in 
his  field.  .  .  .  Finally,  his  whole  array  of  habits  is  swiftly 
obedient  to  serve  in  the  solution  of  new  problems.  Automatism  is 
not  genius,  but  it  is  the  hands  and  feet  of  genius." 

If  these  authors  are  right,  then  it  is  the  first  business  of 
the  teacher  in  Arithmetic  to  see  that  the  right  habits  are 
formed  from  the  beginning  and  that  the  fundamental  habits 
which  enter  into  higher  ones  later  on  shall  be  intelligently 
approached  by  the  teacher.  Certainly,  no  other  branch  of 
school  work  utilizes  more  the  first  habits  in  the  formation 


CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC     197 

of  those  that  follow.  The  child's  first  experience  with  num- 
bers in  school  should  be  made  to  assist  it  in  his  next  step 
and  the  earliest  drills  in  addition  should  be  the  stepping- 
stones  to  higher  functioning  in  the  work  that  follows.  One 
great  fault  has  been  that  schools  have  sought  to  introduce 
children  to  the  mastery  of  problems  involving  situations 
when  the  children  should  have  been  mechanizing  the  pro- 
cesses of  the  fundamentals.  The  preadolescent  period  in 
school  is  distinctively  the  psychological  age  for  drills ;  it  is 
not  the  period  to  demand  either  reasoning  or  imagination 
upon  the  part  of  children.  Hall  (52a,  313)  says:  "Puberty 
is  the  birthday  of  the  imagination,"  and  it  is  discouraging 
and  wasteful  to  thrust  verbal  propositions  upon  young  chil- 
dren, requiring  them  to  wrestle  with  the  interpretation  of  a 
complex  situation,  when  they  should  be  establishing  a  whole 
federation  of  hierarchies  of  mathematical  habits,  so  that  there 
will  be  both  speed  and  accuracy  in  responses  when  there  is 
need  for  them.  We  shall  profit  greatly  if  we  return  to  the 
discarded  oral  Arithmetic,  and  give  patient,  systematic  drills 
upon  rapid  adding,  subtracting,  multiplying  and  dividing, 
without  pencil  or  crayon.  Hundreds  of  examples  should  be 
announced  orally  by  teachers  in  these  four  fundamentals,  and 
the  children  taught  to  respond  quickly.  In  this  way  the  lower 
habits  of  association  are  formed.  With  proper  drills  there 
is  every  reason  to  believe  that  children  in  the  sixth  and  sev- 
enth grades  will  far  surpass  the  present  status  in  those  grades. 
These  rapid  elementary  processes  favor  prompt  fusion  into 
the  higher  unitary  processes,  and  are  also  favorable  for  prompt 
reactions  when  new  emergencies  arise.  Since  one's  response 
to  a  situation  depends  partly  upon  the  rate  of  mental  and 
nervous  processes,  but  "  far  more  upon  how  much  is  in- 
cluded in  each  process,"  it  will  be  seen  how  necessary  it  is 
to  present  only  simple  stimuli  until  they  have  been  mastered. 
In  almost  any  regular  class  in  Arithmetic  can  be  seen  pupils 
who  add,  subtract,  multiply  and  divide  with  great  speed, 
while  others  are  going  along  at  moderate  pace,  and  still  others 
are  laboriously  struggling  with  the  most  elementary  processes 
by  means  of  objective  helps,  like  the  fingers.  Perhaps,  the 
rapid  worker  expends  less  energy  and  finishes  in  one-fourth  the 
time,  because  his  federation  of  habits  has  done  his  work  for 
him,  so  to  speak,  while  the  other  pupils  have  been  going  over 
the  ground  as  if  it  were  strange  territory  to  them.  The  main 
habits  to  be  drilled  into  the  young  arithmetician  are :  ( i )  rapid 
association  of  numbers  in  the  four  fundamental  processes ;  (2) 
accuracy  in  computations;  (3)  alert  and  keen  attention  to 
statement  of  examples  and  problems;  and  (4)  quick  applica- 
tion of  the  proper  process  to  new  situations.    Teachers  should 


198     CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC 

be  extremely  careful  to  avoid  arrested  development  in  the 
lower  order  of  habits, — so  often  seen  in  children  in  the  public 
schools.  By  starting  right,  and  establishing  the  lower  habits 
well,  it  is  easy  to  follow  with  the  proper  sequence  of  simple 
habits,  on  up  into  the  organization  of  the  higher  groups,  or 
hierarchies. 

If  I  am  right  in  placing  the  first  burden  of  responsibility 
upon  the  individual  teacher,  there  is  to  be  an  equal  sharing 
with  the  authors  of  texts  in  Arithmetic.  In  no  other  subject 
in  the  curriculum  does  the  structure  of  the  text-book  deter- 
mine so  largely  the  teacher's  procedure  as  in  Arithmetic.  The 
majority  of  public  school  teachers  follow  closely  the  order  of 
topics  and  the  exact  rules  of  treatment  laid  down  in  the  book. 
Small  change  is  ever  made  in  any  of  the  material  presented. 
It  is  taken  explicitly,  and  no  adaptation  is  made  to  specific 
needs  of  classes  or  individual  pupils. 

It  is  also  true  that  Arithmetic  demands,  more  than  other 
branches,  an  orderly  association,  not  only  of  its  fundamental 
facts,  but  even  more  emphatically,  a  related  and  orderly  group- 
ing of  all  the  subsequent  topics.  It  follows  from  these  de- 
siderata that  the  improvement  in  results  in  Arithmetic  must 
begin  far  back  in  the  treatment  of  the  topics  by  the  authors 
of  series  of  texts  for  school  use.  We  shall  hardly  have  better 
accomplishments  among  pupils  until  our  texts  are  built  upon 
more  psychological  and  pedagogical  lines.  The  present  books 
are  characterized  by  unnecessary  and  indefensible  separation 
of  genetically  related  topics  that  are  given  to  the  children  as 
wholly  new  and  unrelated;  and  as  a  consequence  the  memory 
is  overworked  with  rules,  formulas  and  model  solutions,  and 
there  is  no  systematic  mastery  of  the  topics,  no  orderly  asso- 
ciation of  the  principles  involved,  and  no  formation  of  definite 
habits  of  response.  There  should  be  ample  drills  provided  in 
pure  numbers  for  the  first  four  grades  at  least,  with  emphasis 
upon  the  processes,  and  then  a  gradual  development  into  the 
other  topics  in  such  a  manner  as  to  show  the  genetic  growth 
and  derivation  of  the  later  from  the  earlier  topics. 

VI.    Suggested  Norms  For  Text-Books 

In  order  to  bring  together  the  best  possible  talent  and  ex- 
perience for  the  making  of  texts  in  Arithmetic,  there  is  needed 
the  joint  counsel  of  the  business  world  to  decide  the  most 
necessary  topics  to  be  included,  of  the  pedagogist  to  tell  what 
are  the  practical  reactions  of  children  in  the  school-room,  and 
of  the  psychologist  to  follow  out  the  latest  suggestions  of  sci- 
entific discovery  in  the  laws  of  mental  behavior  and  growth. 

Two  prime  questions  are  met  in  the  preparation  of  the  text- 


CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC      199 

book: — (i)  What  topics  are  to  be  included;  and  (2) how  shall 
they  be  treated, — as  dissociated  from  the  child's  experience 
for  purely  cultural  purposes,  or  as  topics  which  deal  with 
situations  within  the  range  of  the  child's  daily  hfe?  An  ex- 
amination of  the  Arithmetics  used  now  discloses  the  persist- 
ence of  some  topics  from  the  remote  past,  even  despite  the  fact 
that  the  practical  business  world  long  ago  discarded  them  or 
has  modified  them  almost  beyond  recognition.  Partnership 
with  Time  was  inherited  from  the  Hanseatic  League,  which 
made  practical  use  of  it,  but  it  has  remained  in  the  books  for 
generations  after  it  has  been  of  any  service,  as  an  example  of 
the  conservatism  of  tradition.  One  sees  small  reason  for  in- 
cluding in  any  text  for  general  school  use  such  topics  or  sub- 
topics as  the  Vermont  Rule  in  Partial  Payments,  or  even  Par- 
tial Payments  itself,  or  Foreign  Exchange,  Troy  Weight, 
Apothecaries'  Weight,  Surveyor's  Measure,  Brick  Layers* 
Work,  Plastering,  Lumber  Measure,  etc.,  any  more  than  to  put 
in  also  Bakers'  Rules,  Tailoring,  Livery,  Tobacco  Manufac- 
turing, Stock  Raising,  Dress  Making,  and  scores  of  other 
special  lines  in  which  arithmetical  laws  may  be  of  great  ser- 
vice. The  Metric  System  has  no  place  in  the  grades  of  our 
American  schools.  The  entire  school  should  not  be  compelled 
to  study  something  that  only  a  small  percentage  of  the  pupils 
will  ever  get  to  use  in  scientific  work  later.  It  may  well  be 
made  a  part  of  high-school  mathematics  when  it  is  needed. 

Without  further  negative  discussion,  the  following  topics 
are  suggested  as  reasonable  and  adequate  to  the  real  needs  of 
the  public  schools : 

1.  Numeration    and    Notation.     Hindu   and    Roman   both   to   be   pre- 

sented. 

2.  The   Four   Fundamental   Operations,   Subtraction   by   the   Austrian 

and  Division  by  the  Italian  method. 

3.  Tables  of   Denominate   Numbers,   to  include   U.  S.   Money,   Long, 

Dry,    Time,    Liquid,    Avoirdupois,    Square,    Cubic,    and    Circular 
Measures. 

4.  Five   Secondary  Topics, — all  possessing  the  Ratio  Idea. 

(a)  Common  Fractions, — approached  from  the  side  of  Division. 

(b)  Decimal  Fractions, — presented  as  a  special  form  of  common 

fractions  whose  denominator  is  always  some  multiple  of 
10,  the  numerator  alone  being  written  and  the  denom- 
inator indicated  by  the  position  of  the  point. 

(c)  Percentage, — presented  as  a  special  form  of  Decimals,  with 

100  as  its  denominator.  Many  practical  problems  touch- 
ing such  business  lines  as  the  majority  of  people  come  into 
contact  with. 

(d)  Ratio, — which  is  only  a  special   form  of  writing  a  common 

fraction,  placing  a  colon  vertically  between  numerator  and 
denominator  instead  of  a  horizontal  line  between  them  in 
the   usual  manner. 

(e)  Proportion, — simply    an    equality    of    ratios. 


200     CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC 

5.  Square  and  Cube  Root. 

6.  Mensuration:  A  distinct  and  separate  subject  to  be  approached  ob- 

jectively, with  real  objects  and  figures. 

(a)  Rectilinear  Surfaces:  rectangles,  and  triangles. 

(b)  Solids:    including    prisms,    pyramids,    cylinders,    cones    and 

spheres. 

Under  these  six  divisions  the  essential  topics  in  general  have 
been  presented.  An  appendix  might  easily  be  added  to  suit 
particular  communities,  but  it  seems  poor  economy  for  pub- 
lishers and  the  public  to  have  a  burden  of  highly  specialized 
problems  in  the  general  text-book.  In  the  appendix  rural 
communities  could  be  provided  with  rural  problems,  and  those 
confined  to  strictly  factory  communities  with  problems  suitable 
to  their  practical  needs.  It  is  not  to  be  overlooked  that  the 
first  six  grades  especially  are  to  be  provided  with  many  more 
examples  for  drill  purposes  than  are  ordinarily  afforded  now. 
The  Germans  have  always  outdistanced  us  in  this  feature  of 
arithmetical  work.  Their  plan  of  publishing  Hefte  which  con- 
tain from  twenty  to  one  hundred  and  fifty  pages  of  examples, 
and  selling  for  three  to  twelve  cents  each,  has  much  to  com- 
mend itself  to  us.  Hentschel's  Neue  Recheniihel,  for  instance, 
contains  32  pages,  3,760  examples,  and  sold  for  3^  cents  per 
copy.  The  French  have  issued  many  problem  manuals  for 
use  in  their  schools,  selecting  much  material  from  the  official 
examination  lists.  We  can  improve  this  feature  of  our  arith- 
metical work  either  by  adding  special  appendices  when  it  is 
desirable  to  have  the  examples  and  problems  for  drill  in  the 
same  text  with  the  development  of  the  general  theoretical  side, 
and  by  the  issuance  of  separate  booklets  of  drill  exercises  in 
other  instances. 

Examples  and  Problems. — All  arts  and  sciences  have  a  right 
to  their  own  technical  nomenclature.  Their  terms,  of  neces- 
sity, must  be  more  specialized  than  these  terms  are  when  used 
in  the  ordinary  way.  For  this  reason  there  seems  to  be  a  need 
of  making  a  distinction  between  the  exercises  which  deal  only 
with  pure  numbers  and  those  which  apply  numbers  in  stated 
situations.  The  writer  believes  it  would  be  a  great  help  if  all 
text-books  in  Arithmetic  would  refer  to  all  exercises  in  pure 
numbers  as  "Examples,"  and  to  all  exercises  in  applied  num- 
bers as  "Problems."  Both  oral  and  written  exercises  could 
be  given  under  each  one  of  these  classes. 

Algebra  and  Geometry — It  will  not  appear  radical  now  to 
advocate  the  introduction  of  some  elements  of  Algebra  and 
Geometry  into  the  public  school  Arithmetic  text.  In  handling 
numbers  in  the  grammar  grades  the  equation  is  of  much  ser- 


CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC     201 

vice.  In  mensuration  especially  it  is  necessary  to  appeal  to 
Geometry,  as  little  intelligent  work  can  be  done  there  without 
some  knowledge  of  geometrical  figures  and  facts.  In  German 
and  French  schools  use  of  these  subjects  has  been  resorted  to 
for  many  years  to  illuminate  the  processes  in  Arithmetic.  It  is 
not  advisable  to  eliminate  Arithmetic  from  the  7th  and  8th 
grades  in  order  to  introduce  separate  texts  in  Algebra  and 
Geometry.  A  gradual  introduction  of  these  principles  may 
begin  in  the  5th  and  6th  grades,  and  proceed  with  some  definite 
matter  fused  with  the  more  specific  arithmetical  facts.  There 
is  also  an  evident  need  of  continuing  Arithmetic  into,  if  indeed 
not  entirely  through,  the  high-school.  All  the  tests  so  far 
made  show  a  lamentable  weakness  in  Arithmetic  in  the  higher 
grades.  Recently  Stratton  (118,336)  has  emphasized  the  need 
of  sympathy  between  Arithmetic,  and  Geometry  and  Algebra. 
Too  many  teachers  of  secondary  schools  belittle  Arithmetic, 
forgetting  that  the  higher  work  depends  upon  it  While  the 
pupil  has  outgrown  many  of  the  minor  problems  he  must  still 
keep  in  close  touch  with  arithmetical  calculations  in  his  more 
generalized  mathematics.  In  52  schools  reporting  to  the 
American  Committee  of  the  International  Commission  on  the 
Teaching  of  Mathematics,  19  introduce  Algebra  in  the  7th 
and  8th  grades  as  Algebra,  and  3  have  work  in  Geometry 
other  than  what  is  needed  in  mensuration.  In  the  schools  of 
Germany,  Geometry  is  usually  begun  now  in  the  6th  grade. 
Out  of  23,351  departments  reporting  in  England  1,383  chose 
Algebra  as  an  elective.  Geometry  is  taught  in  about  one- 
fourth  as  many  departments.  It  seems  advisable  to  begin 
these  two  subjects  early  as  companion  subjects  along  with 
Arithmetic  and  continue  Arithmetic  into  the  high-school 
work.  For  this  purpose  the  text-books  should  present  all  the 
necessary  material. 


VII.  Some  Suggested  Norms  For  Teaching 

The  many  comparative  experimental  studies  that  have  been 
made  clearly  demonstrate  the  fact  that  good  courses  of  study, 
good  superintendence,  and  well-educated  teachers  with  plenty 
of  equipment  do  not  insure  fruitful  results.  The  Macedonian 
cry  comes  today  from  a  multitude  of  teachers.  Dyer  in  his 
recent  report  to  the  Boston  School  Committee  (38,11)  says: 
"The  immediate  and  most  urgent  need  felt  by  teachers  is 
fresh  light  upon  the  art  of  teaching."  The  writer  has  been 
in  somewhat  close  touch  with  the  rural  schools  in  the  Missis- 
sippi Valley  for  the  past  two  decades  and  he  knows  this  con- 


202     CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC 

dition  that  Dyer  says  exists  in  Boston  is  quite  general, — 
this  one  continuous  entreaty  for  assistance.  We  have  had  an 
affluence  of  theory  and  a  poverty  of  serviceable  help  in  actual 
practice.  Arithmetical  texts  should  not  be  laden  with  peda- 
gogic instruction  to  teachers.  They  need  to  avail  themselves 
of  the  privileges  afforded  in  State  and  City  Normal  schools, 
in  Educational  Departments  in  State  Universities,  in  Teachers' 
Associations  and  Institutes,  in  w^orks  on  pedagogy,  and  in 
school  journals  in  order  to  get  the  most  help  in  methods  of 
teaching.  This  does  not  lessen  the  value  of  good  text-books, 
built  upon  clear,  pedagogical  lines.  Teachers  who  can  really 
teach  Arithmetic, — and  the  other  common  school  subjects, — 
with  life,  with  delight,  throwing  around  each  recitation  the 
best  affective  atmosphere  so  that  every  activity  of  pupil  and 
teacher  will  be  properly  motivated,  are  today  in  great  demand. 
Is  not  the  dread  of  failure  in  school  work  the  worst  kind  of 
failure  ? 

Drill  on  Fundamentals. — Children  enter  school  with  some 
notion  of  number  already  formed,  as  has  been  shown.  The 
teacher's  early  work  with  them  should  be  intelligently  based 
upon  what  they  already  know,  and  should  keep  in  close  touch 
with  the  child's  mode  of  arriving  at  his  first  number-concepts. 
No  book  is  needed.  Abaci,  splints,  counters,  and  cards  with 
large  dots  upon  them  may  be  used.  Together  with  the  pre- 
sentation of  sense-stimuli  to  be  counted  both  simultaneously 
and  in  succession,  the  child  soon  may  be  taught  the  number- 
symbols.  Many  children  know  these  before  entering  school 
and  it  is  well  for  the  teacher  to  lead  gently  the  child's  interests 
rather  than  to  attempt  to  drive  or  force  them.  No  child 
seems  to  be  exclusively  visual-minded,  or  solely  auditory- 
minded,  or  totally  motor-minded,  but  each  child  is  all  of  them 
at  once,  and  may  learn  one  fact  in  one  channel  and  another 
fact  in  another  channel.  So  the  primary  teacher  will  avail 
herself  of  all  the  means  at  hand  to  give  the  concepts  of  num- 
ber in  the  completest  and  happiest  forms,  partly  by  presenta- 
tion, partly  by  counting,  and  many  times  through  the  child's 
own  movements.  As  has  been  earlier  pointed  out,  children 
delight  to  count,  perhaps  because  they  can  do  it.  Like  older 
people  they,  too,  worship  at  the  shrine  of  achievement !  It  is 
also  wise  to  have  them  measure  and  compare  objects  for  basic 
work,  but  it  is  learned  foolishness  and  pedagogic  folly  to  in- 
sist, either  in  texts  in  the  lower  grades  or  in  teaching  Arith- 
metic in  the  grades  that  every  time  one  divides  27  by  9  that  2"/ 
is  necessarily  "  measured  by  the  9  and  is  therefore  concrete." 
When  written  work  is  introduced  it  is  better  to  place  the 
digits  under  each  other  instead  of  in  equation  form  as  is  the 


CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC     203 

practice  so  generally  in  the  first  grade,  since  the  future  ar- 
rangement will  be  with  figures  under  each  other.    Thus, 

146234745961782539 
321453252335116323 

467  687997  12  9689885  12 

These  are  given  merely  as  a  suggestion  on  the  form  in 
which  the  very  first  written  operations  in  addition  should  be 
expressed  and  no  order  of  subject-matter  is  here  attempted. 
The  child  should  be  given  those  forms  and  symbolic  state- 
ments which  he  is  to  use  first  and  new  ones  are  to  be  brought 
in  only  as  he  needs  them.  Arithmetic  should  be  adapted  to 
the  child  and  not  vice-versa.  It  is  best  to  use  written  sub- 
traction in  the  Austrian  form  and  written  division  in  the  long 
form  by  the  Italian  method.  In  all  the  four  fundamental 
operations  constant  and  consistent  drill  is  required  until  the 
mechanical  processes  become  habitual.  If  children  are  to  be 
expert  in  handling  numbers  they  should  be  thoroughly  trained 
in  the  fundamentals  before  they  come  to  the  age  of  puberty. 
All  the  experimental  studies  warrant  this  conclusion.  This 
appears  to  be  the  strategic  time  to  emphasize  work  and  drill 
with  pure  number,  with  examples,  according  to  the  definition 
set  forth  here. 

Why  Are  Problems  Harder  Than  Examples? — In  examples 
the  child  has  his  processes  clearly  indicated  for  him  in  their 
statement.  In  problems,  the  case  is  very  different.  He  has 
to  "  orient "  himself  arithmetically  among  words,  words, 
words,  which  too  frequently  are  scarcely  within  his  compre- 
hension. As  was  mentioned  under  the  experimental  studies, 
marked  variations  have  been  found  in  the  difficulties  of  prob- 
lems by  a  change  of  their  wording.  The  pupil  must  know  the 
meaning  of  all  words  used  in  the  problem  and  also  know  the 
situation  presented,  and  this  demands  that  it  deal  with  objects 
and  situations  familiar  to  the  child.  An  intelligent  and  inde- 
pendent solution  is  possible  only  when  the  language  employed 
is  within  his  vocabulary  and  the  situation  within  his  ex- 
perience. 

Tables  of  Weights  and  Measures. — Early  in  the  child's 
school  experience  the  simpler  measures,  such  as  Long  and 
Liquid  Measures  may  be  begun  objectively,  since  most  chil- 
dren at  school  entrance  have  a  little  knowledge  of  some  of 
the  units,  such  as  foot,  yard,  gallon,  pint,  etc.  All  the  tables 
should  be  approached  as  objectively  as  possible,  so  judgment 
may  precede  and  strengthen  memory.  Longitude  and  Time  is 
practical  now  only  when  the  Standard  Time  belts  are  used. 


204     CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC 

If  it  is  approached  from  the  geographical  side,  with  proper 
drill  upon  the  earth's  rotation,  it  will  not  be  necessary  to  com- 
mit rules  about  ''  when  both  points  are  east "  or  "  when  one 
is  east  of  the  prime  meridian  and  one  is  west/'  since  the  mind 
grasps  the  situation  objectively. 

Fractions,  Percentage,  and  Ratio. — At  school  entrance  most 
children  know  what  is  meant  by  3^,  ^,  and  possibly,  jE4.  Up 
through  the  first  four  grades  the  common  fractions  are  gradu- 
ally employed  as  they  are  needed,  but  in  the  5th  grade  they 
come  in  for  fuller  systematic  treatment.  Decimal  fractions 
should  follow  a  good  understanding  of  common  fractions,  and 
should  always  be  taught  as  a  special  form  of  common  frac- 
tions, which  themselves  should  be  based  upon  Division.  Any 
common  fraction  whose  denominator  is  10,  or  some  multiple 
of  10,  may  be  written  as  a  decimal  by  simply  writing  its  nu- 
merator and  indicating  its  denominator  by  an  agreed  conven- 
tional use  of  the  period  for  a  decimal  point.  Percentage 
should  be  taught  in  very  close  relation  to  decimal  fractions, 
since  it  is  based  upon  the  decimal  whose  denominator  is 
always  100.  If  the  genetic  relation  between  common  and 
decimal  fractions,  and  between  decimals  and  Percentage  is 
clearly  developed  inductively  these  subjects  will  be  vastly 
easier  to  teach  and  to  learn  than  if  they  are  taught  as  separate 
and  unrelated  subjects,  as  is  so  frequently  done. 

Solution  of  Problems. — The  writer  has  found  in  teaching 
Arithmetic  to  several  hundred  Normal  School  students,  the 
majority  of  whom  had  taught  before  entering  his  classes,  three 
distinct  hindrances  to  the  intelligent  solution  of  problems, — 
Rules,  Formulas,  and  Model  Solutions.  If  text-books  give 
rules  or  formulas  or  model  solutions,  students  of  advanced 
experience  even  are  inclined  to  lean  heavily  upon  them.  This 
would  suggest  alertness  upon  the  part  of  teachers  in  the 
grades  to  make  sure  that  children  are  proceeding  intelligently 
and  are  thinking  the  relations  in  the  solutions.  When  a  prob- 
lem is  attacked,  the  first  step  should  be  to  read  it  carefully 
and  pick  out  all  the  things  that  are  granted,  and  to  learn  next 
what  is  to  be  found.  One  set  of  solutions  here  will  serve  to 
exemplify  what  is  meant  and  also  to  correct  some  erroneous 
ideas  that  are  found  still  in  considering  problems  in  Percent- 
age. Keeping  in  mind  all  that  has  been  said  on  the  origin  and 
nature  of  Percentage  and  the  procedure  with  problems  in  gen- 
eral, let  us  take  this  problem: — 

"  If  some  goods  are  sold  at  a  profit  of  12^^%,  for  $1,012.50, 
what  did  they  cost?" 

Smith  (i05,74flf)  in  his  latest  book  on  Arithmetic  uses  this 
problem  to  show  three  erroneous  methods  of  solution,  and 


CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC     205 

decides  that  the  way  out  of  the  difficulty  lies  in  employing  an 
algebraic  symbol  for  the  cost.  He  entirely  overlooks  the  in- 
timate relation  between  Percentage  and  Decimal  Fractions  and 
treats  it  as  a  distinct  part  of  Arithmetic,  and  this  is  a  common 
weakness  of  arithmetical  texts.  His  erroneous  solutions,  so 
far  as  they  are  arithmetical,  should  extend  to  four  and  include 
his  own!  Unnecessary  appeals  to  Algebra  cannot  be  sus- 
tained. When  it  is  needed  to  use  algebraic  or  geometrical 
principles  to  economize  time  and  energy  or  in  heightening  the 
lucidity  of  the  solution,  no  objection  will  be  raised,  but  when 
this  is  made  necessary  because  of  a  lack  of  proper  connection 
between  arithmetical  topics,  then  it  is  to  be  condemned.  If 
the  Hundred-hundredths  base-idea  is  grasped  in  Percentage, 
Smith's  reasoning  will  be  seen  to  deal  with  some  "  straw " 
solutions.  Coming  back  to  the  problem  proposed  for  solution 
we  shall  first  read  it,  and  require  that  the  facts  granted,  and 
the  results  asked  for  shall  be  clearly  set  forth  before  the 
solution  proceeds  and  we  have : 

Granted  (i)   Selling   price   =  $1,012.50 
(2)  Gain  per  cent  ==         12.5 


Required  (3)  Cost  =  what? 


(4)  100% +  12.5%=       112.5% 

Since  (5)  112.5%  of  the  cost  of  the  goods  =  $1,012.50 

Then  (6)  1%      "      "      "      "     «        «       =          ^ 

And  (7)  100%      "      "      "      "     "        "       =       900 

Hence  (8)  The  cost  of  the  goods  =  $900 

The  words  before  the  numbers  of  the  equations  are  given 
merely  to  suggest  the  grouping  of  the  facts  of  the  problem, 
but  are  not  necessary  in  the  pupils*  written  work.  Since  the 
gain  has  been  expressed  in  per  cent  in  the  problem,  it  is  the 
most  natural  procedure  to  take  100%  as  the  basis  in 
the  solution.  When  this  fact  is  seen  the  rest  of  the  work 
is  self-evident.  This  is  in  perfect  harmony  with  such  solu- 
tions as  the  following: 

1.  Find  three-eighths  of  24.  Shall  we  let  5/5  equal  24,  and  then 
reason  ourselves  into  adopting  an  algebraic  symbol  before  we  can 
solve  this?  Evidently,  since  the  cue  has  been  suggested  in  the 
fractional  part,  three-eighths,  the  obvious  plan  is  this : 

Since  8/8  of  24  =  24;  then,  1/8  of  24  ==  3;  and  3/8  of  24  =  9. 

2.  Find  37.5%  of  24. 

Since  100%  of  24  =  24;  then,  1%  of  24  =  .24;  and  37.5%  of  24  =  9. 

3.  15  is  5/8  of  what  number? 

Since  5/8  of  a  number  =  15 ;  then,  1/8  of  it  =  3 ;  and  8/8  of  it  =  24. 

4.  15  is  62.5%  of  what  number? 

Sdnce  62.5%  oJt  a  number  =  15;  then,  1%  of  it  =  .24;  and  100% 

of  it  «=»  24. 


206     CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC 

This  ''  unitary  analysis  "  method  grows  directly  out  of  the 
processes  of  common  fractions,  and  is  to  be  preferred  to  any 
method  that  is  satisfied  to  get  the  ''answer,"  and  cares  nothing 
about  the  relations  of  the  separate  steps  in  the  work.  The 
numbering  of  the  steps  has  been  found  of  some  practical  ser- 
vice m  referring  to  the  solutions,  and  is  not  an  essential  part 
of  the  work, — much  as  the  pages  of  a  book  are  numbered  for 
convenience,  but  have  no  bearing  upon  the  content  of  the  book 
itself. 

Inaccuracies. — Many  text  books  and  more  teachers  are  care- 
less in  arithmetical  statements.  Each  equation  should  be  true 
within  itself  and  should  be  as  carefully  expressed  as  accuracy 
demands.  Arithmetic  is  nothing  if  it  is  not  exact  and  ac- 
curate. One  often  finds  the  continued  equation  in  books, 
somewhat  like  this:  6X  2=^  12-7-4  =  3  X  5  =  I5»  in  which 
the  various  members  are  not  equal.  The  trouble  arises  in  an 
attempt  to  put  two  or  more  separate  steps  into  one  statement. 
While  the  continued  equation  is  sometimes  true  arithmetically, 
it  is  never  correct  grammatically.  The  equality  sign  is  the 
principal  verb,  and  the  second  member  its  object,  which  cannot 
properly  serve  both  as  the  direct  object  of  one  verb  and  be 
the  subject  of  the  next  principal  verb!  Many  authors  also 
continue  to  use  such  expressions  as:  "100%  =$660,"  and 
"Let  100%  =-'$66o,"  entirely  disregarding  the  fact  that  per 
cent  always  means  hundredths  and  that,  therefore,  100%  can 
equal  nothing  but  i  or  some  expression  reducible  to  unity.  To 
say  "100%=? $660  is  to  say  that  i=$66o;  and  to  "Let 
100%  =  $660  is  to  let  I  =  $660,  which  is  too  apparently  false 
to  need  further  mention.  The  %  sign  should  not  be  treated 
as  an  unknown  quantity  in  Algebra.  Yet,  this  fundamental 
error  in  handling  this  simple  matter  has  led  to  endless  con- 
fusion in  the  treatment  of  Percentage  in  the  public  schools. 

Sources  of  Error. — The  mistakes  commonly  found  in  the 
work  of  pupils  in  solving  problems  have  these  sources:  i. 
Mistakes  in  reading  the  problem,  (a)  reversing  figures,  (b) 
not  getting  the  meaning  of  the  words,  or  (c)  not  understand- 
ing the  situations  involved,  with  what  facts  are  granted  and 
what  is  required ;  2.  Inability  to  decide  upon  the  processes  to 
be  used ;  and  3.  Errors  in  computation.  Each  one  of  these 
mistakes  calls  for  a  specific  kind  of  drill  for  its  rectification. 
Careful  work  in  the  lower  grades  can  do  much  to  avoid  them 
in  the  later  work  of  the  children. 

Abstract  and  Concrete  Numbers. — It  is  the  belief  of  the 
writer  that  much  useless  and  wasteful  philosophising  is  at- 
tempted in  the  treatment  of  numbers  in  the  grade  work.     It 


CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC     207 

ought  not  to  be  regarded  as  sound  pedagogy  to  try  to  force 
upon  children  the  introspective  and  ratiocinative  deductions  of 
psychologists  and  philosophers  about  the  nature  of  number. 
One  may  seek  to  discredit  this  position  by  calling  it  an  appeal 
to  the  naive  and  laissez  faire,  but  the  suggestions  have  suffi- 
cient provocation.  In  his  new  book,  Stamper  (113,  29)  would 
almost  confuse  a  philosopher  v/ith  such  statements  as : 

"  Number  is  necessarily  abstract."  "  The  term  abstract  number 
strictly  speaking  expresses  a  redundancy,  for  number  is  essentially 
abstract.  .  .  .  The  abstract  idea  of  number  should  be  consid- 
ered in  multiplication  and  division  problems.  In  multiplication,  the 
multiplier  is  abstract,  the  multiplicand  and  product  being  concrete. 
.     .     .     In  division,  the  dividend  is  always  concrete."       (p.  30.) 

One  cannot  withhold  the  conviction  that  Stamper,  although 
supporting  the  counting  psychosis  as  the  origin  of  number, 
since  he  says  (p.  29)  :  ''  Number  has  its  origin  in  the  counting 
process/'  yields  to  McClellan  and  Dewey  when  they  state 
(76,  137) : 

"  The  multiplicand  must  always  represent  a  number  of  (primary) 
units  of  quantity  (a  measured  quantity),  and  is  commonly  said  to 
be  concrete."  "  From  the  relation  existing  between  multiplication 
and  division,  it  is  seen  that  in  division  the  dividend,— or  multi- 
plicand as  being  a  product  of  two  factors, — always  represents  a  meas- 
ured quantity,  i.  e.,  it  is  concrete;  the  divisor  may  denote  either 
a  concrete  quantity  or  a  pure  number  and  the  quotient  is,  of  course, 
numerical  in  the  one  case  and  interpreted  as  concrete  in  the  other." 

This  philosophical  reasoning  has  played  havoc  in  practical 
school  work,  since  teachers  have  striven  to  foist  it  upon  the 
children  even  in  the  lower  grades.     Allen  well  says  (2,383)  : 

"  If  the  child  sees  that  each  of  seven  boys  has  5  cents,  and  wants 
to  know  how  many  all  have,  he  will  have  little  doubt  as  to  whether 
he  is  repeating  "  5  cents  "  "  seven  boys  times,"  or  "  seven  cents  times  " 
or  just  "seven  times"  and  he  will  need  no  rule  about  abstract  mul- 
tipliers." 

In  other  words,  the  natural  movement  of  mind  in  handling 
these  number  combinations  is  more  certain  and  more  direct,  if 
left  unhampered  by  philosophical  considerations  and  artificial 
rules.    Wark  (131,  162)  claims: 

"  The  most  elementary  counting,  even  that  stage  when  the  counts 
were  not  carried  in  the  mind,  but  merely  in  notches  on  a  stick  or 
by  DeMorgan's  stones  in  a  pot,  requires  some  thought;  and  the 
most  advanced  counting  implies  memory  of  things.  The  terms,  there- 
fore, abstract  and  concrete  number,  have  long  since  ceased  to  be 
used  by  thinking  people." 

Recently  the  writer  visited  an  Arithmetic  class  in  a  State 
Normal  and  saw  a  group  of  practically  adult  students  confused 
about  this  very  question  concerning  abstract  and  concrete  num- 


208     CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC 

bers,  according  to  their  previous  training  in  the  conventionaH- 
ties  of  the  text-book.  Their  teacher  diverted  the  work  of  the 
hour  and  she  and  the  class  spent  almost  the  whole  period  in  re- 
establishing the  requirements  "  that  the  product  must  always 
be  the  same  kind  of  unit  as  the  multiplicand,"  and  "  addends 
mast  all  be  alike  to  be  added."  This  is  not  an  exceptional 
case.  Throughout  the  whole  range  of  teaching  Arithmetic  in 
the  public  schools  pupils  are  obfuscated  by  the  philosophical 
encumbrances  which  have  been  imposed  upon  the  simplest  pro- 
cesses of  numerical  work.  The  time  is  surely  ripe,  now  we 
are  readjusting  our  ideas  of  the  subject  of  Arithmetic,  to  re- 
vise some  of  these  wasteful  and  disheartening  practices.  Al- 
gebra historically  grew  out  of  Arithmetic,  yet  it  has  not  been 
laden  with  this  distinction.  No  pupil  in  Algebra  lets  x  equal 
the  horses;  he  lets  x  equal  the  number  of  horses,  and  pro- 
ceeds to  drop  the  idea  of  horses  out  of  his  consideration.  He 
multiplies,  divides,  and  extracts  the  root  of  the  number,  some- 
times handling  fractions  in  the  process,  and  finally  interprets 
the  result  according  to  the  conditions  of  his  problem.  Of 
course,  in  the  early  number  work  there  have  been  the  sense- 
objects  from  which  number  has  been  perceived,  but  the  mind 
retreats  naturally  from  objectivity  to  the  pure  conception  of 
number,  and  then  to  the  number  symbol.  The  following  is 
taken  from  the  appendix  to  Horn's  thesis  (57),  where  a  7th 
grade  girl  gets  the  population  of  the  United  States  in  1820: 

7,862,166  whites 

233,634  free   negroes 
1,538,022  slaves 


9,633,822 


In  this  problem  three  different  kinds  of  addends  are  combined, 
if  we  accept  the  usual  distinctions.  Some  may  say  that  this 
is  a  mistake, — that  the  pupil  transformed  the  "  whites,"  ''  free 
negroes  "  and  "  slaves  "  into  a  common  unit,  such  as  "  people  " 
or  "  population  "  and  then  added  these  common  units.  But 
this  "  explanation  "  is  entirely  gratuitous,  as  one  will  find  if 
he  questions  the  pupil  about  the  process.  It  will  be  found  that 
the  child  simply  added  the  figures  as  numbers  only  and  then 
interpreted  the  result,  according  to  the  statement  of  the  prob- 
lem, without  so  much  mental  gymnastics.  The  writer  has 
questioned  hundreds  of  students  in  Normal  School  work  on 
this  point,  and  he  believes  that  the  ordinary  mind-movement 
is  correctly  set  forth  here,  no  matter  how  well  one  may  main- 
tain as  an  academic  proposition  that  this  is  not  logical.  Many 
classes  in  the  Eastern  Kentucky  State  Normal  have  been  given 
this  problem  to  solve,  and  they  invariably  get  the  same  result : 


CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC     209 

"  In  a  garden  on  the  Summit  are  as  many  cabbage-heads  as  the 
total  number  of  ladies  and  gentlemen  in  this  class.  How  many 
cabbage-heads  in  the  garden?" 

And  the  black-board  solution  looks  like  this  each  time: 

29  ladies 

15  gentlemen 

44  cabbage-heads 

So,  also  one  may  say:  I  have  6  times  as  many  sheep  as  you 
have  cows.  If  you  have  5  cows,  how  many  sheep  have  I? 
Here  we  would  multiply  the  number  of  cows,  which  is  5,  by  6 
and  call  the  result  30,  which  must  be  linked  with  the  idea  of 
sheep  because  the  conditions  imposed  by  the  problem  demand 
it.  The  mind  naturally  in  this  work  separates  the  pure  num- 
ber from  its  situation,  as  in  Algebra,  handles  it  according  to 
the  laws  governing  arithmetical  combinations,  and  labels  the 
result  as  the  statement  of  the  problem  demands.  This  is  ex- 
pressed in  the  following,  which  is  tacitly  accepted  in  Algebra, 
and  should  be  accepted  equally  in  Arithmetic : 

In  all  computations  and  operations  in  Arithmetic,  all  numbers  are 
essentially  abstract  and  should  be  so  treated.  They  are  concrete 
only  in  the  thought  process  that  attends  the  operation  and  interprets 
the  result. 

Mental,  or  Oral,  Arithmetic. — Much  oral  drill  should  be 
given  in  all  the  first  six  grades.  Processes  should  become 
almost  automatic  by  the  close  of  these  grades.  The  German 
schools  excel  in  Kopfrechnen,  and  our  American  schools  can 
well  afford  to  return  to  the  practice  of  a  generation  ago  in  this 
matter. 

The  writer  does  not  think  that  he  has  spoken  the  last  word 
for  improvements  needed  in  teaching  and  in  studying  Arith- 
metric,  but  he  does  believe  that  he  has  pointed  to  some  reliefs 
from  the  present  discouragements  in  the  results.  There  is  no 
gainsaying  his  position  that,  in  the  ultimate  analysis,  the  fail- 
ure to  respond  quickly  and  correctly  to  problems  in  Arith- 
metric  lies  in  the  mental  inability  of  the  child.  And  beyond 
this  fact  we  must  remember  that  the  child  must  do  his  own 
thinking  in  the  subject.  No  superintendent,  no  teacher,  no 
course  of  study,  no  text-book  can  do  more  than  provide  him 
with  the  proper  stimuli  and  direction  for  his  own  develop- 
ment. Growth  must  come  from  within.  Tests  and  experi- 
mental studies,  such  as  have  been  brought  together  in  this  dis- 
sertation, will  assist  in  discovering  to  us  the  child's  reactions 
to  numbers  and  to  problems  involving  numbers,  and  in  this 
way  serve  to  adapt  the  means  of  the  school  to  the  ends  of 


210     CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC 

individual  education.  A  sensible  synthesis  of  all  that  has  been 
found  good  in  these  studies,  and  that  may  yet  be  disclosed  in 
similar  investigations,  will  hasten  the  Renaissance  in  Arith- 
metic. 

VIII.  General  Summary 

1.  Arithmetic  is  one  of  the  chief  subjects  in  the  first  eight 
grades  of  the  schools  throughout  the  civilized  world.  It  re- 
ceives about  one-sixth  of  the  total  school  time. 

2.  Children  show  some  interest  in  numbers  about  the  close 
of  their  fourth  year,  and  at  school  entrance  at  six  they  have 
quite  well  formed  ideas  of  numbers  up  to  8  or  lo.  Many  are 
able  to  count  lOO  or  more. 

3.  One  school  of  educators  holds  that  the  number  concept 
arises  from  simultaneously  perceived  groups  of  objects; 
another  school  holds  that  number  is  gained  from  successively 
perceived  stimuli.  One  would  teach  by  presenting  many  ob- 
jects in  a  group;  the  other  would  teach  by  counting. 

4.  No  definite  correlation  has  been  found  to  exist  between 
the  number  systems  employed  by  primitive  peoples  and  their 
civil  development.  Number  systems  are  known  to  arise  readily 
when  these  are  needed. 

5.  Both  primitive  peoples  and  children  have  a  natural  ten- 
dency to  symbolisms  and  this  fact  makes  the  growth  of  the 
number  symbols  easy  for  them. 

6.  One  group  of  educators  advocates  deferring  number  work 
until  one  or  two  grades  have  been  completed  by  the  child  in 
school.  Another  group  favors  the  usual  practice  of  starting 
children  in  numbers  as  soon  as  they  enter  school  at  six  years 
of  age.  The  child's  pre-school  interest  in  numbers  would  sup- 
port the  latter  group,  while  the  successful  number  work  re- 
ported by  the  English  Infant  Schools  and  by  Montessori  is 
also  favorable  to  this  view. 

7.  The  influence  of  puberty  upon  arithmetical  interests  and 
abilities  has  been  found  to  be  very  marked,  and  it  should  be 
clearly  understood  by  teachers  who  handle  children  of  the 
postpubertal  period. 

8.  Drill  work  has  been  found  exceedingly  valuable  for  de- 
veloping skill  in  handling  numbers  rapidly  and  accurately. 
Short  periods  are  better  than  the  same  total  time  used  in  long 
periods  of  drill.  The  permanent  effect  of  drill-work  has  been 
found  to  be  much  greater  than  was  supposed. 

9.  Children  are  inclined  to  make  certain  habitual  "  type- 
errors."  These  should  be  carefully  listed  by  teachers  and 
made  the  subject  of  vigorous  drill.  Much  time  is  squandered 
in  going  over  matter  which  the  children  already  know  in  most 


CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC     211 

schools,  and  energy  is  not  concentrated  upon  the  particular 
difficulties  that  should  be  attacked. 

10.  If  problems  in  Arithmetic  are  made  to  deal  with  situa- 
tions within  the  experience  of  pupils,  the  subject  will  be 
robbed  of  much  of  its  present  tedium. 

11.  The  adoption  of  a  '*  fixed  standard  "  for  achievement  in 
Arithmetic,  rather  than  a  progressive  one,  threatens  to  work 
great  harm  in  the  schools.  The  criterion  by  which  to  judge 
the  results  of  teaching  should  be  a  growing  one,  or  stagna- 
tion will  be  encouraged. 

12.  The  wording  of  problems,  which  are  to  be  distinguished 
from  examples,  should  receive  careful  attention  in  the  text- 
books. 

13.  Great  individual  differences  exist  among  children  con- 
cerning Arithmetic.  These  variations  range  from  dread  aver- 
sion to  the  subject  all  the  way  to  special  favoritism  for  it. 
Early  impressions  received  from  parents,  tutors,  associates, 
and  school  experiences  all  seem  to  have  some  influence  in 
determining  the  Einstellung  of  the  pupils  toward  Arithmetic. 

14.  Girls  have  been  found  to  excel  boys  generally  in  prac- 
tical skill  in  computations,  but  boys  are  better  than  girls  in 
reasoning  upon  number  situations. 

15.  Children  in  the  first  five  grades  do  not  have  the  range 
of  attention  or  the  power  of  concentration  that  the  children 
in  the  upper  grades  possess.  The  dawn  of  adolescence  marks 
the  dividing  line  between  the  two  types. 

16.  Habits  of  rapid  and  accurate  handling  of  numbers  must 
be  thoroughly  established  in  the  first  five  grades.  Drill  to  this 
end  should  be  begun  as  early  as  possible  after  school  entrance, 
and  carried  forward  with  special  care  in  the  3rd,  4th,  and  5th 
grades.  Habits  of  combining  numbers  quickly  should  be 
firmly  fixed  and  hierarchies  of  habits  should  respond  auto- 
matically to  given  processes,  while  whole  federations  of  hier- 
archies should  gather  spontaneously  to  solve  questions  in- 
volving several  processes.  Much  drill  should  be  afforded  with 
pure  numbers,  and  a  minimum  of  problems  involving  a  study 
of  situations  resorted  to. 

17.  Problems  may  be  begun  in  the  6th  grade  with  profit, 
and  the  work  will  be  attractive  and  inviting  if  the  pupils  can 
perform  the  operations  with  ease. 

18.  Philosophical  distinctions  between  abstract  and  concrete 
numbers  are  not  to  be  urged  upon  the  children  in  the  ele- 
mentary schools.  However  true  these  distinctions  are  in  logic, 
to  teach  them  in  the  grades  serves  no  useful  or  needful  end, 
and  adds  to  the  burden  of  dealing  with  practical  applications 
of  number. 

19.  Oral,  or  mental.  Arithmetic  deserves  a  wide  place  in  the 


212     CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC 

early  grades,  and  should  not  be  omitted  even  from  the  upper 
grades.  Kopfrechnen  has  a  value  that  pencil  and  crayon  can 
never  supplant. 

20.  Instead  of  teaching  Arithmetic  in  the  eight  grades  of  the 
elementary  school,  Algebra  in  the  first  two  grades  of  the  high 
school  and  Geometry  in  the  last  two  high  school  grades,  it 
would  be  better  from  several  considerations  to  teach  Ele- 
mentary Mathematics  through  the  entire  twelve  grades,  intro- 
ducing Algebra  and  Geometry  as  early  as  they  can  be  of  real 
service,  and  continuing  Arithmetic  into  the  high  school. 

21.  Text-books  in  Arithmetic  should  have  the  combined  ex- 
perience and  knowledge  of  the  business  man,  the  psychologist, 
and  the  pedagogist.  If  the  authorship  of  texts  combines  these 
three  interests,  a  much  more  practical  and  teachable  book  will 
result. 

22.  The  derivative  relation  of  Common  Fractions,  Ratio 
and  Proportion  to  Division;  of  Decimals  to  Common  Frac- 
tions; and  of  Percentage  to  Decimals  should  be  clearly  set 
forth  in  the  text-books,  beyond  which  so  few  teachers  are  dis- 
posed to  go  in  their  teaching.  This  change  in  the  books 
would  do  more  than  any  one  other  thing  to  clarify  the  topics, 
which  stand  now  as  apparently  separate  and  unrelated  topics 
to  be  treated  independently. 


IX.  Bibliography 

1.  Aley,   R.  J.  and  McNiell,  I.  C.     Mathematics  in  the  Grades. 

N.  E.  A.  Report,  1908,  pp.  569-576. 

2.  Allen,    Fiske.      Is    Arithmetic   a    Science   of    Numbers   or   of 

Symbols?      Kansas    School    Magazine,    vol.    i,    no.    9,    Nov., 
IQ12,  pp.  378-384. 

3.  Alling-Aber,   Mary   R.     An   Experiment  in   Education.     New 

York,  Harpers,   1897,  245  p. 

4.  Andrews,    W.    S.     Magic   Squares.     Monist,   vol.    15,   pp.   429- 

461,  554-586.     Magic  Cubes.     Monist,  vol.  16,  pp.  328ff. 

5.  Arnett,  L.   D.     Counting  and  Adding.     Amer.  Jour.   Psychol., 

vol.  16,  no.  3,  July,  1905,  pp.  327-336. 

6.  Badanes,  Saul.     The  Falsity  of  the  Grube  Method  of  Teach- 

ing Primary  Arithmetic.     New  York  University  thesis,  1895, 

47  P- 

7.  Bailey,   M.  A.  and  Cook,  John  W.     The  Teaching  of  Arith- 

metic  in    Elementary   Grades.     N.    E.   A.   Report,    18^5,   pp. 

380-387. 

8.  Ball,  W.  W.  R.     A  Short  Account  of  the  History  of  Mathe- 

matics.    London,   Macmillan,   1888,  464  p. 

9.  Ballard,    P.    B.     The   Teaching   of    Mathematics    in    London 

Public  Elementary  Schools.     Ed.  Dept.,  Great  Britain  Special 
Rep.,  vol.  26. 
ID.    Boas,    Franz.     Language   and   Thought.      Introduction   to   the 
Handbook    of    American     Indian     Languages.      Washington, 
Gov.  Print.  Office,  191 1,  1069  p. 


CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC     213 

11.  Branford,    Benchara.     a    Study   of    Mathematical    Education, 

including  the  Teaching  of  Arithmetic.  Oxford,  Clarendon 
Press,  1908,  392  p. 

12.  Brooks,    S.    D.     Annual    Report    of   the    Superintendent,   July, 

1910,  School  Document  No.  10,  1910,  Boston  Public  Schools. 

13.  Brown,  J.   C.     An  Investigation  on  the  Value  of   Drill  Work 

in  the  Fundamental  Operations  in  Arithmetic.  Jbur.  of 
Ed.  Psychol.,  vol.  2,  Feb.,  191 1,  81-88;  vol.  3,  Nov.,  1912, 
485-492;  Dec,  1912,  561-570. 

14.  Brown,    Wiluam.     The    Psychology   of     Mathematics.     Child 

Study,  vol.  6,  March  and  April,  1913,  24-26;  42-47. 

15.  Browne,    C.    E.     The    Psychology  of   the   Simple   Arithmetical 

Processes.     Am.  Jour.  Psychol.,  vol.  17,  1906,  1-37. 

16.  Bruce,  H.  A.  B.     Bending  the  Twig.    American  Magazine,  vol. 

69,  1 910,  690-695. 
i6a.  Bryan,    W.   L.    and    Harter,    N.     The   Telegraphic   Language. 
Psych.  Rev.,  vol.  6,  no.  4,  July,  1899,  345-375- 

17.  BuRNHAM,  W.  H.    Arithmetic  and  School  Hygiene.    Ped.  Sem., 

vol.  18,  191 1,  54-73. 

18.  Cajori,  Florian.    a  History  of  Mathematics.    New  York,  Mac- 

millan,  1894,  422  p. 

19.  Calfee,  J.  E.     Rural  Arithmetic.     Boston,  Ginn  and  Co.,  1913, 

119  p. 

20.  Campbell,  C.  V.    Courses  in  Arithmetic.    N.  Y.  Teachers  Mono- 

graphs, 1912,  28-34. 

21.  Carmichael,  R.  D.    a  Lesson  from  the  History  of  Numbers. 

School  Sci.  and  Math.,  vol.  13,  no.  5,  May,  1913,  392-399- 

22.  Chartres,  W.  W.    Teaching  the   Common   Branches.     Boston, 

Mifflin,  1913,  355  p. 

23.  Cole,  L.  W.     Adding  Upward  and  Downward.    Jour.  Ed.  Psy- 

chol, vol.  3,  Feb.,   1912,  83-94. 
23a.  CoLBURN,     Warren.       Teaching     of     Arithmetic.       Elementary 
School  Teacher,  vol.  12,  June,  191 2,  pp.  463-480. 

24.  CoNANT,  L.  L.     Historical  Development  of  Arithmetical  Nota- 

tion.    Ped.  Sem.,  vol.  2,  149-152. 

25. .    The  Number  Concept.     New  York,  Macmillan,  1896, 

218  p. 

26.  Cosby,  Byron.     Arithmetical  Teaching, — An  Experiment  in  the 

Mound  City,  Mo.,  Public  Schools.  School  Sci.  and  Math., 
vol.  II,  191 1,  629-634. 

27.  Courtis,  S.  A.     Standard  Tests  in  Arithmetic.     Jour.  Ed.  Psy- 

chol., vol.  2,  191 1,  272-274. 

28. .     Standard  Scores  in  Arithmetic.     Elementary  School 

Teacher,  vol.   12,   Nov.,   1912,   127-137. 

29. .  Measurement  of  Growth  and  Efficiency  in  Arith- 
metic. Elementary  School  Teacher,  vol.  10,  Oct.,  1909,  58-74; 
Dec,    1909,    177-199;    vol.    II,    Dec,    1910,    171-185;    March, 

1911,  360-370;  June,   1911,  528-539. 

30. .    Report  on   the   Courtis   Tests  in  the   City  of   New 

York,  1911-1912.  Interim  Report,  Committee  on  School 
Inquiry,  Board  of  Estimate  and  Apportionment.     1913,  158  p. 

31. .    Bulletin    No.    2,    Courtis    Standard   Tests.     Detroit, 

Aug.,   1913,  44  p. 

^2.  Dawson,  N.  H.  R.  Simplification  of  Arithmetic.  U.  S.  Com. 
Ed.  Rep.,  1886-7,  238-239. 

32a.  Dearborn,  G.  V.  N.  Moto-Sensory  Development.  Baltimore, 
Warwick  &  York,  1910,  215  p. 


214     CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC 

33a.  Decroly,  Dr.  M.  le,  and  Degand,  Mlle.  Julia.  L'evolution  des 
notions  de  quantites  continues  et  discontinues  chez  L'enfant. 
Arch.   d.   Psych.,  vol.    12,   19 12,  pp.  81- 121. 

33.  DeMorgan,   Augustus.     Arithmetical   Books    from   the   Inven- 

tion of  Printing  down  to  the  Present  Time.     Lx)ndon,  Tay- 
lor and  Walton,   1847,   124  p. 

34.  — ' —  — ' — '.    On    the    Study    and    Difficulties     of     Mathematics. 

Chicago,  Open  Court  Pub.  Co.,  1898,  288  p. 

35.  Deuchler,    G.     Psychologische   Vorfragen   des   ersten   Rechen- 

unterrichts.     Zeits.  f.  pad.  Psych.,  1912,  vol.  13,  36-52. 

36.  DoLBEAR,  Katherine  E.     Prccocious  Children.     Ped.  Sem.,  vol. 

19,  no.  4,  Dec,  1912,  461-491. 

37.  Dooley,  W.  H.    Arithmetic  in  a  Massachusetts  Industrial  School. 

School   Sci.   and  Math.,  vol.   11,   191 1,  246-249. 

38.  Dyer,  F.   B.     Annual  Report  of  The  Superintendent.     Boston, 

Dec.  15,  1913,  244  p. 

39.  Eliot,    Chas.    W.     Shortening   and    Enriching    the    Grammar 

School  Course.     N.  E.  A.  Rep.,  1892,  617-620. 

40.  Erler,   Otto.     Zahlengedachtnis  und   Rechenfertigkeit.     Zeit.   f. 

pad.  Psych.,  vol.  12,  1911,294-295. 
40a.  Freeman,   F.    N.     Uber    den   Aufmerksamkeitsumfang    u.    die 
Zahlauffasung,    Leipzig,  Hahn,  1910,  pp.  88-168. 

41.  Galton,   Francis.     Inquiries   into  Human   Faculty  and   Devel- 

opment.    London,  Macmillan,  1883,  387  p. 

42.  Gartland,  p.  G.    Two  Experiments  on  Grammar  School  Gradu- 

ates.    School  Sci.  and  Math.,  vol.   11,  1911,  I55-I59- 

43.  Gilbert,  C.  B.     What  Children  Study  and  Why.     Boston,  Sil- 

ver, Burdett  and  Co.,   1913,  331   p. 

44.  Gildemeister,  Thelda.    Mathematics  in  the  Elementary  Schools 

of  the  United  States.     U.   S.  Bureau  Ed.  Bui.,   191 1,  whole 
no.  460,  75-120. 

45.  Giles,  J.  T.    The  Scientific  Study  of  Arithmetic  Work  in  School. 

N.  E.  A.  Rep.,   1912,  488-492. 

46.  Gow,  James.     A  Short  History  of  Greek  Mathematics.     Cam- 

bridge, University  Press,  1884,  323  p. 

47.  Greenwood,  J.   M.     Verbatim   Report  of   Recitations  in  Arith- 

metic and  Language  in  the  Schools  of  Kansas  City,  Mo.,  U.  S. 
Com.  Ed.  Rep.,  1893-4,  vol.  i,  557-594- 

48. .    Report   of    the    Committee   of    Fifteen — Dissent   on 

Arithmetic.     U.  S.  Com.  Ed.  Rep.,   1893-4,  vol.   i,  532-4;  or 
N.  E.  A.  Rep.,  1895.  333- 

49.  and  Martin,  A.  Notes  on  the  History  of  American  Text- 
books on  Arithmetic.  U.  S.  Com.  Ed.  Rep.,  1897-8,  vol.  i, 
789-868;   1898-9,  vol.  I,  781-837. 

50.  Griggs,  A.  O.     Pedagogy  of  Mathematics.     Ped.  Sem.,  vol.   19, 

1912,  350-375- 

51.  Hall,  F.   H.   and  Gilbert,   N.   D.     Imagination  in  Arithmetic. 

N.  E.  A.  Rep.,  1897,  62iff. 

52.  Hall,  G.  Stanley.     Educational  Problems,  New  York,  Apple- 

ton,    1911,   2   vols.,    1424   p.      See    "The    Pedagogy   of    Ele- 
mentary Mathematics,"  vol.  2,  341-396. 
52a. .    Adolescence.      New    York,    Appletons,    1904,    vol.    i, 

589  p. 

53.  Harris,   W.   T.     Report   of  the  Committee  of   Fifteen.     U.    S. 

Com.   Ed.   Rep.,   1893-4,   vol.   i,  497-502;   or  N.   E.   A.   Rep., 
1895,  296-301. 


CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC     215 

54.  Harrison,   A.    S.     The   Demonstration    School   Record   No.   2. 

The  Pursuits  of  the  Fielden  School,  Manchester,  Eng.  Uni- 
versity Press,   1913,  275  p. 

55.  Hart,  W.  W.     Community  Arithmetic  for  Seventh  and  Eighth 

Grades.     Elementary  School  Teacher,  vol.   11,  285-295. 

56. — .    Mathematics    in    the    Elementary     Schools    of     the 

United  States.  U.  S.  Bureau  Ed.  Bui.  No.  13,  191 1,  whole 
no.  460,  16-65. 

57.  Horn,    Ernest.      Experiment    in     Seventh     Grade    Arithmetic. 

Univ.  of  Mo.,  master's  thesis,  1908.  Typewritten,  69  p.  with 
appendix.      (Borrowed  from   Univ.  of   Mo.   Library.) 

58.  Hornbrook,  A.  R.    An  Open  Letter  to  Students  Teaching  Arith- 

metic in  the  Training  Department.    San  Jose,  Gal,  1913,  20  p. 

59.  Jackson,    L.    L.     The    Educational    Significance    of    Sixteenth 

Century  Arithmetic.  New  York,  Teachers  College,  190^ 
232  p. 

60.  Janicke,  E.     Geschichte    der    Methodik    des    Rechenunterrichts, 

vol.   3,   Kehr's   Geschichte,   Gotha,   Thiermann,   1888. 

61.  Jeffrey,  G.  R.     Some  Observations  on  the  Use  of  the  "  Reck- 

oning Tests."     Jour.  Exp.  Ped.,  vol.  i,  1911-12,  392-396. 

62.  JosEiPHUS,    Flavius.     Antiquities    of    the   Jews.      Philadelphia, 

Porter,  1829,  978  p. 

63.  JuDD,  C.  H.     Studies  in  Principles  of  Education.     Initiative,  or 

the  Discovery  of  Problems.  Elementary  School  Teacher, 
vol.   13,   1912-3,   I46flf. 

64.  KiKUCHi,  B.   D.     Japanese  Education.     London,  Murray,   1909, 

397  P- 

65.  KiRBY,  T.  J.     Practice  in  the  Case  of   School  Children.     New 

York,  Teachers  College,   1913,  98  p. 

66.  Knilling,  R.    Die  Naturgemasse  Methode  des  Rechenunterrichts 

in  der  Deutschen  Volksschule.  Munich,  Oldenbourg,  1897-9, 
2  vols.,  372  p.  and  266  p. 

67.  Lay,  W.  a.    Fiihrer  durch  den  ersten  Rechenunterrichts  nature- 

gemasses  Lehrverfahren  gegriindet  auf  psychologische  Ver- 
suche  und  angeschlossen  an  die  Entwicklungsgeschichte  des 
Rechenunterrichts.     Karlesruhe,  Nemnich,   i89i8,  155  p. 

68.    and   Enderlin,   M.     Fiihrer  durch   das   erste   Schuljahre. 

Leipzig,  Quelle,  191 1,  434  p. 

69.    ' .    Alte   und    Neue    Experimente   zum    ersten    Rechen- 

unterricht.     Zeits.  f.  exp.  Pad.,  Bd.  i,   129-166. 

70.  Lessing,    T.      Padagogik    und     Psychologie     der    Mathematik. 

Zeits.  f.  exp.  Pad.,  Bd.  9,  225-237. 

71.  Lewis,  E.  O.     Popular  and  Unpopular  School  Subjects.    Jour. 

Exp.  Ped.,  June,  1913,  vol.  2,  no.  2,  89-98. 

72.  LiETZMANN,  W.     Stoflf  und  Methode  des  Rechenunterrichts  in 

Deutschland.     Leipzig,  Teubner,  1912,  107  p. 

73.  LoBSiEN,     M.       Korrelation     zwischen     Zahlenged^chtnlis     und 

Rechenleistung.     Zeits.  f.  exp.  Pad.,  Bd.  12,  54-60. 

74.  Lucas,  Edouard.     Theorie  des  Nombres.     Paris,  Gauthier,  1891, 

520  p. 
74a.  Maday,  Stefan  v.     Die  Fahigkeit  des  Rechnens  beim  Menschen 
und   beim   Tiere.     Zeits.    f.    angewandte    Psychol,    u.    psych. 
Sammelforschung,   Bd.  8,  Heft,  3  U-  4,  PP-  204-227. 

75.  Major,  D.  R.    First  Steps  in  Mental  Growth.    New  York,  Mac- 

millan,  1896,  360  p. 


216     CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC 

76.  McLellan,  J.  A.  and  Dewey,  John.  The  Psychology  of  Num- 
ber.    New   York,   Appleton,   1895,  309  p. 

TJ.  Mecker,  H.  H.  a  Study  in  Seventh  Grade  Subject  Matter. 
Thesis,  1913,  Univ.  of  Mo.,  typewritten,  84  p.  (Borrowed 
from  U.  of  Mo.  Library.) 

78.  Messenger,   J.    F.     The   Perception   of    Number.     New   York, 

Macmillan,  1903,  44  p. 

79.  Meumann,   Ernst.     Vorlesungen   zur   Einfiihrung  in   die   Ex- 

perimentelle  Padagogik  und  ihre  Psychologischen  Grundlagen. 
Leipzig,   Englemann,   1907,   2  vols. 

80. .    Ueber    die    Anlage    zum    Rechnen.      Zeits.    f.    exp. 

Pad.,  Jahr.  14,  Feb.,  1913,  117-118. 

81.  Mill,  J.  S.     Demonstration  and  Necessary  Truths.     A  System 

of  Logic,  bk.  2,  ch.  6,  New  York,  Longmans,  1900. 

82.  MiLLis,  J.  F.    The  Solution  of  Problems.    Kansas  School  Mag., 

vol.  2,  no.  9,  Nov.,  1913,  305-310. 

83.  MoNTESSORi,  M.     The  Montessori  Method.     New  York,  Stokes, 

1912,  377  p.     See  chap.  9,  "Teaching  of  Numeration;  Intro- 
duction to  Arithmetic." 

84.  MiJNSTERBURG,  HuGO.    Psychology  and  the  Teacher.    New  York, 

Appletons,  1909,  330  p. 

85.  Myers,  C.  S.     Introduction  to  Experimental  Psychology.    Cam- 

bridge, Univ.   Pr.,   191 1,   156  p. 

86.  Nanu,  Helene  A.    Zur  Psychologie  der  Zahlauffasung.    Wiirz- 

burg,  Becker,  1904,  56  p. 

87.  Newcomb,  Simon  and  Aley,  R.  J.    The  Teaching  of  Arithmetic. 

N.  E.  A.  Rep.,  1906,  86-102. 

88.  Paine,  Cassie  L.     A  Strong  Motivation  for  Arithmetic  Work. 

Elementary    School    Teadher,    vol.    13,    April,    1913,    379-386. 

89.  Palmer,    G.    W.     The    Teaching   of    Arithmetic  in    Secondary 

Schools.     Great   Britain   Special   Ed.   Rep.,  vol.   26,   224-256. 

90.  Payne,  B.  R.    Public  Elementary  School  Curricula.    New  York, 

Silver,   Burdett  and  Co.,   1905,  200  p. 

91.  Perry,  A.  C.    Problems  of  the  Elementary  School.    New  York, 

Appletons,  1910,  224  p. 

92.  Phelps,  C.  L.     A  Study  of  Errors  in  Tests  in  Adding  Ability. 

Elementary  School  Teacher,  vol.  16,  no.  i,  Sept.,  1913,  29-39. 

93.  Phillips,   F.   M.     Value  of   Daily   Drill  in   Arithmetic.     Jour. 

Ed.   Psychol.,  vol.  4,  no.  3,  March,  1913,   159-163. 

94.  Philups,  D.  E.    The  Genesis  of  Number  Forms.    Amer.  Jour. 

Psychol,  vol.  8,  no.  4,  July,  1897,  506-527. 

95. .    Number    and   its   Application.     Ped.    Sem.,    vol.    5, 

no.  2,  Oct.,  1897,  221-281. 

96.  Preyer,  W.     The  Mind  of  the  Child.     New  York,  Appletons, 

1889,  317  p. 

97.  Ranschburg,  p.     Zur  physiologischen  und  pathologischen  Psy- 

chologie  der  elementaren   Rechenarten.     Zeits.   f.   exp.    Pad., 
Bd.  7,  135-162,  1908;  Bd.  9,  1909,  251-263. 

98.  Rice,  J.  M.     Scientific  Management  in  Education.     New  York, 

Publishers  Printing  Co.,  1913,  282  p. 

99.  Rusk,  R.  R.    Introduction  to  Experimental  Education.    London, 

Longmans,  1912,  303  p. 

100.  Santerre,  S.  Psychologie  du  Nombre  et  des  Operations  ele- 
mentaire  de  L'Arithmetique.     Paris,  Doin,  1907.     178  p. 

loi.  Scripture,  E.  W.  Arithmetical  Prodigies.  Aaner.  Jour.  Psy- 
chol., vol.  4,  no.  I,  also  reprint,  Worcester,  Mass.,  1891, 
l^  p. 


CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC     217 

loia.  ScHANOFF,  BoTju.     Die  Vorgange  des  Rechnens.     Padagogische 
Monographien,    Bd.   XL,   Leipzig,    191 1,    120  p. 

102.  Shaw,  O.  A.    Arithmetic  Philosophically  Taught,  or  a  Descrip- 

tion of  the  Visible  Numerator.     Boston,  Marvin,  1832,  95  p. 

103.  Smith,  A.   C.     The  Teaching  of  Arithmetic.     School  Sci.  and 

Math.,   June,   19 12,  vol.    12,  457-460. 

104.  Smith,  D.  E.    The  Teaching  of  Elementary  Mathematics.    New- 

York,  Macmillan,  1901,  312  p. 
105. .    The    Teaching    of    Arithmetic.      Boston,    Ginn    and 

Co.,  1913,  196  p. 
106. .    Early    Drill   in   Arithmetic.     Kansas    School   Mag., 

vol.  I,  no.  3,  March,  1912,  95-101. 
107. .    Article  on  "Arithmetic"  in  Monroe's  Cyclopedia  of 

Education.      New    York,    Macmillan,    191 1,   5    vols.;    vol.    i, 

203-207. 
108.    .    The  International  Commission  on  the  Teaching  of 

Mathematics.    Ed.  Rev.,  vol.  45,  January,  1913,  1-7. 

109. .     Sex  in  Mathematics.     Ed.  Rev.,  vol.  9,  1895,  84-88. 

no.     Sonnenschein,   a.     The   Study   of   Arithmetic  in   Elementary 

Schools.     Great    Britain    Spec.    Rep.    on    Ed.    Subj.,    vol.   8, 

1902,  571-535. 

111.  Smith,  W.  H.    All  the  Children  of  All  the  People.    New  York, 

Macmillan,  1912,  335  p. 

112.  Spencer,  H.  J.    The  Teaching  of  Mathematics  in  the  English 

Public    Elementary    Schools.      Great    Brit.    Spec.    Ed.    Rep., 
vol.  26,  1912,  31-60. 

113.  Stamper,  A.  W.    A  Text-book  on  the  Teaching  of  Arithmetic. 

New  York,   American  Book  Company,   1913,  284  p. 

114.  Starch,  Daniel.    Transfer  of  Training  in  Arithmetical  Opera- 

tions.   Jour.  Ed.  Psychol,  vol.  2,  306-310. 

115.  Stephens,    Irene.     The   Teaching   of    Mathematics   to    Young 

Children.     Great  Brit.  Sp.  Ed.  Rep.,  vol.  26,  1912,  61-79. 

116.  Stitt,  E.  W.    School  and  Business  Arithmetic.    N.  E.  A.  Rep., 

1900,  566-572. 

117.  Stone,  C.  W.     Arithmetical  Abilities  and  Some  Factors  Deter- 

mining Them.    New  York,  Teachers  College,  1908,  102  p. 
118. .     Problems  in  the  Scientific  Study  of  Teaching  Arith- 
metic.   Jour.  Ed.  Psychol.,  January,  1913,  vol.  4,  no.  i,  1-16. 

119.  Stone,  J.  C.     The  Modernization  of  Arithmetic.     N.  E.  Jour. 

Ed.,  vol.  78,  Dec.  4,  I9I3»  573-578. 

120.  Stratton,   W.  F.     The  New  Thought  in  Arithmetic.     Kansas 

School  Mag.,  Dec,  1913,  vol.  2,  336-7. 

121.  Sully,  James.     Studies  of  Childhood.     New  York,  Appletons, 
I  1896,  527  p. 

122.  SuzzALO,  Henry.     The  Teachmg  of  Primary  Arithmetic.     Bos- 

ton, Mifflin,  1911,  124  p. 

123.  Tear,  J.  H.  and  Cox,  H.  C.     The  New  Arithmetic.     N.  E.  A. 

Rep.,  1897,  630-6. 

124.  Thoma,   F.   J.     Lay's   "Fuhrer   durch   den   Rechenunterrichts " 

und  das  Rechnen  auf  Grund  der  gegliederten  Reihe.     Zeits. 
f.  exp.  Pad.,  Bd.   11,  1910,  15-23. 

125.  Thorndike,  E.  L.     Effect  of  Practice  in  the  Case  of  a  Purely 

Intellectual   Function.     Amer.   Jour.    Psychol.,   vol.    19,   July, 

1908,  374-384. 
126. .     Practice   in    the    Case    of    Addition.      Amer.    Jour. 

Psychol.,  vol.   21,   July,   1910,  483-6. 
127. .    The  Measurement  of  Educational  Products.     School 

Rev.,  vol.  20,  no.  5,  May,  1912,  289-309. 


218     CONTRIBUTION  TO  THE  PEDAGOGY  OF  ARITHMETIC 

128. .  Educational  Psychology,  vol.  2.  New  York,  Teach- 
ers College,  191 3,  452  p.  See  chap.  7.  "Amount,  Rate  and 
Limit  of  Improvement." 

129.  Tropfke,    J.      Geschichte    der    Elementar-Mathematik    in    sys- 

tematischen    Darstellung.      Erst.    Bd. :    Rechnen    u.    Algebra, 
Leipzig,   Veit,    1902,   332  p. 

130.  VoiGT,   W.     Ueber  die  Anlage   zum   Rechnen.     Arch.   f.   Pad., 

2.  teil,  vol.  I,  1912,  129  ff. 

131.  Wark,  Anna  L.    Early  Work  in  Number.    Chap,  in  Education 

by  Life,  edited  by  Henrietta  Brown  Smith,  London,   Philip, 

1912,  211  p. 

132.  Webb,  H.  E.    The  Re-introduction  of  Arithmetic  into  the  High 

School  Course.     School  Sci.  and  Math.,  vol.  13,  no.  6,  June, 

1913,  517-524. 

133.  Wells,  F.  L.     The  Relation  of  Practice  to  Individual  Differ- 

ences.    Amer.  Jour.   Psychol.,  vol.  23,  no.   i,  January,   1912, 
75-88. 

134.  Whitley,   M.   T.     An   Empirical   Study  of   Certain   Tests   for 

Individual   Differences.     Arch,   of   Psychol,  no.    19,   August, 
191 1,  146  p. 

135.  Williams,  T.  A.     Intellectual   Precocity.     Ped   Sem.,   vol.   18, 

March,  19 11,  85-103. 

136.  Winch,  W.  H.     Mental  Adaptation  during  the  School  Day  as 

Measured   by   Arithmetical    Reasoning.     Jour.    Ed.    Psychol, 
Jan.,  1913,  vol.  4,  no.  i,  17-28;  Feb.,  1913  ,vol.  4,  no.  2,  71-84. 

137.  '     .    Accuracy  in   School   Children:     Does   Improvement 

in    Numerical    Accuracy   "Transfer?"     Jour.    Ed.    Psychol, 
vol.  I,  1910,  557-589;  vol  2,  1911,  262-271  and  334-336. 

138.  YocuM,  A.  D.     An  Inquiry  into  the  Teaching  of  Addition  and 

Subtraction.     Philadelphia,  Avil  Print.  Co.,  1901,  92  p. 
139. .    A  First  Step  in  Inductive  Research  into  the  Most 

Effective    Methods    of    Teaching    Mathematics.      School    Sci. 

and  Math.,  vol.  13,  no.  3,  March,  1913,  197-210. 
140.    Young,  J.  W.  A.     The  Teaching  of  Mathematics.     New  York, 

Longmans,  1907,  351  p. 
141. .    The  Fifth  International  Congress  of  Mathematicians. 

School  Sci.  and  Math.,  vol.  12,  1912,  702-715. 


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